{
"query": {
"display": "tangent of $$f\\left(x\\right)=\\sec\\left(x\\right)+\\tan\\left(x\\right)$$",
"symbolab_question": "PRE_CALC#tangent f(x)=\\sec(x)+\\tan(x)"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Derivative Applications",
"subTopic": "Tangent",
"default": "y=-\\frac{1}{-1+\\sin(a_{0})}x+\\sec(a_{0})+\\tan(a_{0})+\\frac{a_{0}}{-1+\\sin(a_{0})}",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "Tangent line to $$f\\left(x\\right)=\\sec\\left(x\\right)+\\tan\\left(x\\right):{\\quad}y=-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}x+\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}$$",
"steps": [
{
"type": "step",
"primary": "Compute the tangent line to the general point $$x=a_{0}$$"
},
{
"type": "interim",
"title": "Find the tangent point:$${\\quad}\\left(a_{0},\\:\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)\\right)$$",
"steps": [
{
"type": "step",
"primary": "Plug $$x=a_{0}$$ into the equation $$f\\left(x\\right)=\\sec\\left(x\\right)+\\tan\\left(x\\right)$$",
"result": "f\\left(x\\right)=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)"
}
],
"meta": {
"interimType": "Tangent Find Tangent Point Title 0Eq"
}
},
{
"type": "interim",
"title": "Find the slope of $$f\\left(x\\right)=\\sec\\left(x\\right)+\\tan\\left(x\\right):{\\quad}\\frac{df\\left(x\\right)}{dx}=-\\frac{1}{-1+\\sin\\left(x\\right)}$$",
"input": "f\\left(x\\right)=\\sec\\left(x\\right)+\\tan\\left(x\\right)",
"steps": [
{
"type": "step",
"primary": "In order to find the slope of the function, take the derivative of $$\\sec\\left(x\\right)+\\tan\\left(x\\right)$$"
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(\\sec\\left(x\\right)+\\tan\\left(x\\right)\\right)=-\\frac{1}{-1+\\sin\\left(x\\right)}$$",
"input": "\\frac{d}{dx}\\left(\\sec\\left(x\\right)+\\tan\\left(x\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\frac{d}{dx}\\left(\\sec\\left(x\\right)\\right)+\\frac{d}{dx}\\left(\\tan\\left(x\\right)\\right)"
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(\\sec\\left(x\\right)\\right)=\\sec\\left(x\\right)\\tan\\left(x\\right)$$",
"input": "\\frac{d}{dx}\\left(\\sec\\left(x\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\sec\\left(x\\right)\\right)=\\sec\\left(x\\right)\\tan\\left(x\\right)$$",
"result": "=\\sec\\left(x\\right)\\tan\\left(x\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhUUF3UJORQ8DF3WZn5zWtf8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcauh1OQVXcgjWTXKHGPsynBG6jeh7+jKEzLb7VNCEMF3Z/EO7HD6hbLz/aXlXb+GmzHlg3H3uceHRj0/DC6rcPNEPLgpVAJxpZk+NFSCyCsLpL"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(\\tan\\left(x\\right)\\right)=\\sec^{2}\\left(x\\right)$$",
"input": "\\frac{d}{dx}\\left(\\tan\\left(x\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\tan\\left(x\\right)\\right)=\\sec^{2}\\left(x\\right)$$",
"result": "=\\sec^{2}\\left(x\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYvcxE7pKCugNaT0kVoj74lf8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaulj+bk6iGNhD/yCLteK/hgXNWyGcX6HZt1LGXH2QGa+L9zETukoK6A1pPSRWiPviV0n0gSTsS3EeHwZguA6FoaAkt3WiGR7ZaCaXvz77bMjS"
}
},
{
"type": "step",
"result": "=\\sec\\left(x\\right)\\tan\\left(x\\right)+\\sec^{2}\\left(x\\right)"
},
{
"type": "interim",
"title": "Simplify $$\\sec\\left(x\\right)\\tan\\left(x\\right)+\\sec^{2}\\left(x\\right):{\\quad}-\\frac{1}{-1+\\sin\\left(x\\right)}$$",
"input": "\\sec\\left(x\\right)\\tan\\left(x\\right)+\\sec^{2}\\left(x\\right)",
"result": "=-\\frac{1}{-1+\\sin\\left(x\\right)}",
"steps": [
{
"type": "interim",
"title": "Express with sin, cos",
"input": "\\sec^{2}\\left(x\\right)+\\sec\\left(x\\right)\\tan\\left(x\\right)",
"result": "=\\frac{1+\\sin\\left(x\\right)}{\\cos^{2}\\left(x\\right)}",
"steps": [
{
"type": "step",
"primary": "Use the basic trigonometric identity: $$\\sec\\left(x\\right)=\\frac{1}{\\cos\\left(x\\right)}$$",
"result": "=\\left(\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}+\\frac{1}{\\cos\\left(x\\right)}\\tan\\left(x\\right)"
},
{
"type": "step",
"primary": "Use the basic trigonometric identity: $$\\tan\\left(x\\right)=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$",
"result": "=\\left(\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}+\\frac{1}{\\cos\\left(x\\right)}\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}"
},
{
"type": "interim",
"title": "Simplify $$\\left(\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}+\\frac{1}{\\cos\\left(x\\right)}\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}:{\\quad}\\frac{1+\\sin\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$",
"input": "\\left(\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}+\\frac{1}{\\cos\\left(x\\right)}\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}",
"result": "=\\frac{1+\\sin\\left(x\\right)}{\\cos^{2}\\left(x\\right)}",
"steps": [
{
"type": "interim",
"title": "$$\\left(\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}=\\frac{1}{\\cos^{2}\\left(x\\right)}$$",
"input": "\\left(\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$",
"result": "=\\frac{1^{2}}{\\cos^{2}\\left(x\\right)}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Apply rule $$1^{a}=1$$",
"secondary": [
"$$1^{2}=1$$"
],
"result": "=\\frac{1}{\\cos^{2}\\left(x\\right)}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Aj2A6AXyYKc8onjZ0sgaWzmJIjHHZfwsEbPoUxhGnibdd47a0hQ8flDbGsI5To1doDDfDl9rb93jWiqTfsVP2oyn+6RwSLXV5OKcL4lbWczxKuqsNbiYVwQr6E82pk3yhVRZd12QVEnoVlrOlIz53hcePqTC4e0/dg4Du22gww41tGDKu4M6fVRgkPA6fHYbsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "interim",
"title": "$$\\frac{1}{\\cos\\left(x\\right)}\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\frac{\\sin\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$",
"input": "\\frac{1}{\\cos\\left(x\\right)}\\cdot\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$",
"result": "=\\frac{1\\cdot\\:\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(x\\right)}"
},
{
"type": "step",
"primary": "Multiply: $$1\\cdot\\:\\sin\\left(x\\right)=\\sin\\left(x\\right)$$",
"result": "=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(x\\right)}"
},
{
"type": "interim",
"title": "$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)$$",
"input": "\\cos\\left(x\\right)\\cos\\left(x\\right)",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$\\cos\\left(x\\right)\\cos\\left(x\\right)=\\:\\cos^{1+1}\\left(x\\right)$$"
],
"result": "=\\cos^{1+1}\\left(x\\right)",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$1+1=2$$",
"result": "=\\cos^{2}\\left(x\\right)"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgJyajQjkqczngxvLtluw47oN3fOm5Kcpc0NdzQFiDj9ovYKijQYhJDCbxu/nAOJVxXBxD1gYRAlNp97nQuTZFXRu5R8U1G8Rh9s+llHwfqtic1bCnH3jLV3vr22vWk8gIJE6eFSdaQPkT4FMktmcw=="
}
},
{
"type": "step",
"result": "=\\frac{\\sin\\left(x\\right)}{\\cos^{2}\\left(x\\right)}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s743+Hzh+1NcELvtHIMKV2UJpcm25bt01x735HjJbPvo3qgZ8ivCY+EPo3Waoy7TA3vgODWgpJNtyK6jgKaIhT2KuO77lnnV79llIPeBEfCMF8P1Lyfqcqv55BNoa+zvyv5v7SDswKC5gKvBLFgGVe7v8//6/nV5O4fb8Xgwi7maoM1C0oTTDNuG5cHg9kgIjYbzp+8xc4332N5qNLjjfyda9olvnPgOe7yrqN22AA906pTBAhBmDPYgFnwicq9YO5JUlWCZjg569zb7lkFxZgPduF28nSBS0vr9kA0SNwoz4="
}
},
{
"type": "step",
"result": "=\\frac{1}{\\cos^{2}\\left(x\\right)}+\\frac{\\sin\\left(x\\right)}{\\cos^{2}\\left(x\\right)}"
},
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$",
"result": "=\\frac{1+\\sin\\left(x\\right)}{\\cos^{2}\\left(x\\right)}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
}
],
"meta": {
"interimType": "Trig Express Sin Cos 0Eq"
}
},
{
"type": "interim",
"title": "Rewrite using trig identities",
"input": "\\frac{1+\\sin\\left(x\\right)}{\\cos^{2}\\left(x\\right)}",
"result": "=-\\frac{1}{-1+\\sin\\left(x\\right)}",
"steps": [
{
"type": "step",
"primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$",
"secondary": [
"$$\\cos^{2}\\left(x\\right)=1-\\sin^{2}\\left(x\\right)$$"
],
"result": "=\\frac{1+\\sin\\left(x\\right)}{1-\\sin^{2}\\left(x\\right)}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{1+\\sin\\left(x\\right)}{1-\\sin^{2}\\left(x\\right)}:{\\quad}-\\frac{1}{\\sin\\left(x\\right)-1}$$",
"input": "\\frac{1+\\sin\\left(x\\right)}{1-\\sin^{2}\\left(x\\right)}",
"result": "=-\\frac{1}{\\sin\\left(x\\right)-1}",
"steps": [
{
"type": "interim",
"title": "Factor $$1-\\sin^{2}\\left(x\\right):{\\quad}-\\left(\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(x\\right)-1\\right)$$",
"input": "1-\\sin^{2}\\left(x\\right)",
"result": "=-\\frac{1+\\sin\\left(x\\right)}{\\left(\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(x\\right)-1\\right)}",
"steps": [
{
"type": "step",
"primary": "Factor out common term $$-1$$",
"result": "=-\\left(\\sin^{2}\\left(x\\right)-1\\right)",
"meta": {
"practiceLink": "/practice/factoring-practice",
"practiceTopic": "Factoring"
}
},
{
"type": "interim",
"title": "Factor $$\\sin^{2}\\left(x\\right)-1:{\\quad}\\left(\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(x\\right)-1\\right)$$",
"input": "\\sin^{2}\\left(x\\right)-1",
"steps": [
{
"type": "step",
"primary": "Rewrite $$1$$ as $$1^{2}$$",
"result": "=\\sin^{2}\\left(x\\right)-1^{2}"
},
{
"type": "step",
"primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$",
"secondary": [
"$$\\sin^{2}\\left(x\\right)-1^{2}=\\left(\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(x\\right)-1\\right)$$"
],
"result": "=\\left(\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(x\\right)-1\\right)",
"meta": {
"practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares",
"practiceTopic": "Factor Difference of Squares"
}
}
],
"meta": {
"interimType": "Algebraic Manipulation Factor Title 1Eq"
}
},
{
"type": "step",
"result": "=-\\left(\\sin\\left(x\\right)+1\\right)\\left(\\sin\\left(x\\right)-1\\right)"
}
],
"meta": {
"interimType": "Algebraic Manipulation Factor Title 1Eq"
}
},
{
"type": "step",
"primary": "Cancel the common factor: $$1+\\sin\\left(x\\right)$$",
"result": "=-\\frac{1}{\\sin\\left(x\\right)-1}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ErAtmynW3ylOCv8l2LYToJkLJx4Rwrt2TsPzhPuRRxU8h+u5fV4iBAkECPyXf7J5zMFYmi1F5Hg/ibpEToVnYzzWb8LoTKqv9AUhejSex0THKFLeM1O2mxfgXsXcnng/TeQKHeh69S6dnv9vSoUoFIEBzU9p8/iJr5pwRbtu+vlDjKkqgBstUF8ZuUhzBzcOA/cYyEe1LWypruxvTHwZlQ=="
}
}
],
"meta": {
"interimType": "Trig Rewrite Using Trig identities 0Eq",
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}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+ulNvawWu0thKETU1ikYdsx2B8K/7A1qOW6xZpL5zKItOtZYwUjyXhDTsNnn6ElryOsg4xTbsj8PJfnagYu7Q0byXHpZiAvjzNhrc8StefQOvbl8PATd1Pti7BfR9u+p7kAjP76qW66lOUsURwT0nbgzVXQ2YPdrzdkpGTOw6rdpCyREYXT7vvXrK7OD3maPjAU1tbPq2GdamIthauqJig=="
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "-\\frac{1}{-1+\\sin\\left(x\\right)}"
}
],
"meta": {
"interimType": "Slope Equation Top 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWSwieLOu0rR4+1ppbmqk1MCMmNlMuZmRdJIJqi03IswWETeQKHeh69S6dnv9vSoUoFNhEAUjge2w+ZGLMk7E66nEUwqeq267vW8wTAulNCqDLm/0p8RvkeKnuL7J24BTQ1g=="
}
},
{
"type": "interim",
"title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}$$",
"steps": [
{
"type": "step",
"primary": "Plug $$x=a_{0}$$ into the equation $$-\\frac{1}{-1+\\sin\\left(x\\right)}$$",
"result": "-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}"
}
],
"meta": {
"interimType": "General Equation Slope At Point 2Eq"
}
},
{
"type": "interim",
"title": "Find the line with slope m=$$-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}$$ and passing through $$\\left(a_{0},\\:\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)\\right):{\\quad}y=-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}x+\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}$$",
"steps": [
{
"type": "step",
"primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}$$ and passing through $$\\left(a_{0},\\:\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)\\right)$$"
},
{
"type": "interim",
"title": "Compute the $$y$$ intercept:$${\\quad}b=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}$$",
"steps": [
{
"type": "step",
"primary": "Plug the slope $$-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}$$ into $$y=mx+b$$",
"result": "y=\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)x+b"
},
{
"type": "step",
"primary": "Plug in $$\\left(a_{0},\\:\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)\\right)$$: $$\\quad\\:x=a_{0},\\:y=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)$$",
"result": "\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)=\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b"
},
{
"type": "step",
"primary": "Isolate $$b$$"
},
{
"type": "interim",
"title": "$$\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)=\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b{\\quad:\\quad}b=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}$$",
"input": "\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)=\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b",
"steps": [
{
"type": "step",
"primary": "Switch sides",
"result": "\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)"
},
{
"type": "interim",
"title": "Move $$\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}\\:$$to the right side",
"input": "\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)",
"result": "b=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}",
"steps": [
{
"type": "step",
"primary": "Subtract $$\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}$$ from both sides",
"result": "\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}"
},
{
"type": "interim",
"title": "Simplify",
"input": "\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}",
"result": "b=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}",
"steps": [
{
"type": "interim",
"title": "Simplify $$\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}:{\\quad}b$$",
"input": "\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}+b-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}",
"steps": [
{
"type": "step",
"primary": "Add similar elements: $$\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}=0$$"
},
{
"type": "step",
"result": "=b"
}
],
"meta": {
"interimType": "Generic Simplify Specific 1Eq"
}
},
{
"type": "interim",
"title": "Simplify $$\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}:{\\quad}\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}$$",
"input": "\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)-\\left(-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}\\right)a_{0}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$-\\left(-a\\right)=a$$",
"result": "=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{1}{-1+\\sin\\left(a_{0}\\right)}a_{0}"
},
{
"type": "interim",
"title": "$$\\frac{1}{-1+\\sin\\left(a_{0}\\right)}a_{0}=\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}$$",
"input": "\\frac{1}{-1+\\sin\\left(a_{0}\\right)}a_{0}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:a_{0}}{-1+\\sin\\left(a_{0}\\right)}"
},
{
"type": "step",
"primary": "Multiply: $$1\\cdot\\:a_{0}=a_{0}$$",
"result": "=\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DG1vM/l0c2wGa2A36HJ9O4/1KoYm+IUqu38hk/+97DR8kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcaup+q3oLXvqJ/nE8tEJXJTJrXkntgeDSendyhC2udvITrB5NXi7bTNbXVXk8s7Avpa9rF8pfOdE6vjpLxg61ggGVwPWJE7525HVtzwjml1FdjOz4uqs3WC93/gdCD/TC2KoCnXNf8Pn9MUpUXJjcDB50="
}
},
{
"type": "step",
"result": "=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{\\sin\\left(a_{0}\\right)-1}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xiPY0eqdzaPxeQuxchWzK5nzRB1n4t5S/VURXy0uNOyAUF0N/B0in2Qe4UosENTteWuwrhVA/iXPdLMFaxFJkVXTSum/z5kLpMzXS1UJIeyDWpdMLT0XBdBF3SNHh+8636/LD/rEK6Mn7RXq5z3S0vSeANHT07ZJ5KwtFElbRz65LboCjAWVBumHlB7B42hBFghhMDU/Loy7l4foTVciWB4pgUWEah0lniZLlD4X0wsSFjOzzd81gdwn9WJrsGp5v37UrAOnx+4D3IwRgdwgLwBM7PI3gWYuZR01nMcyXR5LjitDbZuQuWpXLDba7LayialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "step",
"result": "b=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}"
}
],
"meta": {
"interimType": "Generic Simplify 0Eq"
}
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Equations"
}
},
{
"type": "step",
"result": "b=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}"
}
],
"meta": {
"interimType": "Line Equation Find Intersection From Point 0Eq"
}
},
{
"type": "step",
"primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}$$ and $$\\mathbf{b}=\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}$$",
"result": "y=-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}x+\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}"
}
],
"meta": {
"interimType": "Line Equation Slope Point 6Eq"
}
},
{
"type": "step",
"result": "y=-\\frac{1}{-1+\\sin\\left(a_{0}\\right)}x+\\sec\\left(a_{0}\\right)+\\tan\\left(a_{0}\\right)+\\frac{a_{0}}{-1+\\sin\\left(a_{0}\\right)}"
}
],
"meta": {
"solvingClass": "PreCalc"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "tangent f(x)=\\sec(x)+\\tan(x)"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solution
tangent of
Solution
Solution steps
Compute the tangent line to the general point
Find the tangent point:
Find the slope of
Find the line with slope m= and passing through