sin^2(x)tan^2(x)=tan^2(x)+sin^2(x)

{ "query": { "display": "$$\\sin^{2}\\left(x\\right)\\tan^{2}\\left(x\\right)=\\tan^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "symbolab_question": "EQUATION#\\sin^{2}(x)\\tan^{2}(x)=\\tan^{2}(x)+\\sin^{2}(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)\\tan^{2}\\left(x\\right)=\\tan^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right){\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin^{2}\\left(x\\right)\\tan^{2}\\left(x\\right)=\\tan^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\tan^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)$$ from both sides", "result": "\\sin^{2}\\left(x\\right)\\tan^{2}\\left(x\\right)-\\tan^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Express with sin, cos", "input": "-\\sin^{2}\\left(x\\right)-\\tan^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)\\tan^{2}\\left(x\\right)", "result": "\\frac{-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)-\\cos^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}=0", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\tan\\left(x\\right)=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=-\\sin^{2}\\left(x\\right)-\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}+\\sin^{2}\\left(x\\right)\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}" }, { "type": "interim", "title": "Simplify $$-\\sin^{2}\\left(x\\right)-\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}+\\sin^{2}\\left(x\\right)\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}:{\\quad}\\frac{-\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "input": "-\\sin^{2}\\left(x\\right)-\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}+\\sin^{2}\\left(x\\right)\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}", "result": "=\\frac{-\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}=\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "input": "\\left(\\frac{\\sin\\left(x\\right)}{\\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{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73yQ3tP4bkGLx3qZ5P//Et+wNnQzv5E758+LKB991QDZZ8mEc9fB7wfq6et5j4rXP/aL2Coo0GISQwm8bv5wDiY41upviZdufqq2kqjE8aV7m/tIOzAoLmAq8EsWAZV7u/z//r+dXk7h9vxeDCLuZqn2Nii2xSdcID3T1P97D5/LLPC6zR1wBiPFMG2LthZ1XxPJzd9XLwQ6NF+TMQiN1CMlvOh0Yt5E8Wt4FYC70B67bhdvJ0gUtL6/ZANEjcKM+" } }, { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}=\\frac{\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "input": "\\sin^{2}\\left(x\\right)\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}\\right)^{2}=\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "input": "\\left(\\frac{\\sin\\left(x\\right)}{\\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{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73yQ3tP4bkGLx3qZ5P//Et+wNnQzv5E758+LKB991QDZZ8mEc9fB7wfq6et5j4rXP/aL2Coo0GISQwm8bv5wDiY41upviZdufqq2kqjE8aV7m/tIOzAoLmAq8EsWAZV7u/z//r+dXk7h9vxeDCLuZqn2Nii2xSdcID3T1P97D5/LLPC6zR1wBiPFMG2LthZ1XxPJzd9XLwQ6NF+TMQiN1CMlvOh0Yt5E8Wt4FYC70B67bhdvJ0gUtL6/ZANEjcKM+" } }, { "type": "step", "result": "=\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}\\sin^{2}\\left(x\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sin^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" }, { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)=\\sin^{4}\\left(x\\right)$$", "input": "\\sin^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\sin^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)=\\:\\sin^{2+2}\\left(x\\right)$$" ], "result": "=\\sin^{2+2}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+2=4$$", "result": "=\\sin^{4}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sl7YIc5jltiBLO8+U/drtUpK+bfM6jx4Ye9egSi89K/NGoPE9TME3q+OPmgkv2RQCmmh+usYAjSZ4PpsrKEj4Djm97MBerKIQEUtRBdiE/hzH4TSUX465btHtZB1V+p7/WWL+YLPFnDKbll7bITWYYpN4KGI/dKJ9pxbnejSc4U=" } }, { "type": "step", "result": "=\\frac{\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WDPnC+ypjLLtPxhpGKwmnuQu7VFEhOcx4fUG560OJho8Gr62AaVQeXZkDYSAfdAuVdNK6b/PmQukzNdLVQkh7NMQ0vmOWo9IZfhei7w2gJb+UHkj8oFGu50O0FVI/VMBDJOz5IJoi2ceAVm90FrwR/ae3DzsVDZzEjALKmoX1uL3/p4LIWKiC/0/8EWx5YLhozl0bcR5dG+ckm9kDsjMxktk/ha2CrmOaiOEkKURybxMEkNJKaZwoiycOsxK5+MBIG4R2TUpC6+adjN8mKyjrw==" } }, { "type": "step", "result": "=-\\sin^{2}\\left(x\\right)-\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}+\\frac{\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" }, { "type": "interim", "title": "Combine the fractions $$-\\frac{\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}+\\frac{\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}:{\\quad}\\frac{-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "result": "=-\\sin^{2}\\left(x\\right)+\\frac{\\sin^{4}\\left(x\\right)-\\sin^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Convert element to fraction: $$\\sin^{2}\\left(x\\right)=\\frac{\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "result": "=-\\frac{\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}+\\frac{-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)+\\sin^{4}\\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": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)-\\cos^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "Factor $$-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)-\\cos^{2}\\left(x\\right)\\sin^{2}\\left(x\\right):{\\quad}-\\sin^{2}\\left(x\\right)\\left(1-\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)\\right)$$", "input": "-\\sin^{2}\\left(x\\right)+\\sin^{4}\\left(x\\right)-\\cos^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\sin^{4}\\left(x\\right)=\\sin^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)$$" ], "result": "=-\\sin^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)\\sin^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)\\cos^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$-\\sin^{2}\\left(x\\right)$$", "result": "=-\\sin^{2}\\left(x\\right)\\left(1-\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "-\\sin^{2}\\left(x\\right)\\left(1-\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin^{2}\\left(x\\right)=0\\lor\\:1-\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=0$$", "result": "x=0+2πn,\\:x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=0+2πn,\\:x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$1-\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+πn$$", "input": "1-\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "1-\\sin^{2}\\left(x\\right)+\\cos^{2}\\left(x\\right)", "result": "1+\\cos\\left(2x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)=\\cos\\left(2x\\right)$$", "result": "=1+\\cos\\left(2x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Or05b1t1zjl/mxWcXarjrGDuL/+6ie9rzeVbiKZ0CzTaj2VUeWgC00zqTCXgfBdNgGFMQkUTMNnEnFExdO5KvDLzJz4rkYnjv1/W3glakSm1s1RQOlcS2PC6wMP3eHirscKSvc1lJ2BbI3clVxmjBWRLd2VwIqlBNByF6663sySYtHHZIJ8Nhz40VaXBzMWWyHImFc42gKJRGjZFTjbQQUaaUzYDBaJ7PWzJE8yB6KA=" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "1+\\cos\\left(2x\\right)=0", "result": "\\cos\\left(2x\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "1+\\cos\\left(2x\\right)-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(2x\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(2x\\right)=-1$$", "result": "2x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "2x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$2x=π+2πn:{\\quad}x=\\frac{π}{2}+πn$$", "input": "2x=π+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=π+2πn", "result": "x=\\frac{π}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{π}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{π}{2}+πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7s+VDSy9Nl5n0ZrAJ3ZUcLvdVtrzNWwcLBNloZMNuEPDya/gmXRIc30tREPIIyab8RoVJA0H1fnxI+eKaoiWe2vVBR0+98CcmmVtl3U0SxOdktDoFNMUm3jC8/4NQOpcsvFZh+W/iDMDAo4iCI7tFUdwpqaFhGel8eqIiHymEdPNIrutd8eSMgUo+HpRyFwAMvyWHSiFDzdj0zFeH9Fmo6pJrjOMK2FuGAOmiXS/q84KyymwjYXuOrsKGT27ODRXtoZYhIcgpXWG6eZE/QGld7YY0gpHtq/lmoGhPFJQVeUEGNbMh/G+sezf4P31vCapYHSlSpqlNtjpO1BeRthdUJg0ujtYBLJ1WC80jJsZXqFmWd6dPs4XZ31bvn9AQqOpXN+KC5+3NJ4c1zO8G6nx0iFn1Vd9tJZhqnKZRbDSW3wk0kWmJ5SrBIzXSaSVmLePIpppnx5/axq/YTRqF0w/ris7pnHUPzfvWHvpbR3pn3x0Z0XiSDb9NeVHrQfJw/ralufGs4ObHx3NlJ/ofxGIIlghTQ7NPxL7UwD055yztwZbn/kYd/GhR0QG7qH14yEAw" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{π}{2}+πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{2}+πn" }, { "type": "step", "primary": "Since the equation is undefined for:$${\\quad}\\frac{π}{2}+πn$$", "result": "x=2πn,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin^{2}(x)\\tan^{2}(x)-\\tan^{2}(x)-\\sin^{2}(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }