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Popular Trigonometry >

2cot(x)+tan(x)=-2

  • Pre Algebra
  • Algebra
  • Pre Calculus
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Solution

2cot(x)+tan(x)=−2

Solution

NoSolutionforx∈R
Solution steps
2cot(x)+tan(x)=−2
Subtract −2 from both sides2cot(x)+tan(x)+2=0
Rewrite using trig identities
2+tan(x)+2cot(x)
Use the basic trigonometric identity: tan(x)=cot(x)1​=2+cot(x)1​+2cot(x)
2+cot(x)1​+2cot(x)=0
Solve by substitution
2+cot(x)1​+2cot(x)=0
Let: cot(x)=u2+u1​+2u=0
2+u1​+2u=0:u=−21​+i21​,u=−21​−i21​
2+u1​+2u=0
Multiply both sides by u
2+u1​+2u=0
Multiply both sides by u2u+u1​u+2uu=0⋅u
Simplify
2u+u1​u+2uu=0⋅u
Simplify u1​u:1
u1​u
Multiply fractions: a⋅cb​=ca⋅b​=u1⋅u​
Cancel the common factor: u=1
Simplify 2uu:2u2
2uu
Apply exponent rule: ab⋅ac=ab+cuu=u1+1=2u1+1
Add the numbers: 1+1=2=2u2
Simplify 0⋅u:0
0⋅u
Apply rule 0⋅a=0=0
2u+1+2u2=0
2u+1+2u2=0
2u+1+2u2=0
Solve 2u+1+2u2=0:u=−21​+i21​,u=−21​−i21​
2u+1+2u2=0
Write in the standard form ax2+bx+c=02u2+2u+1=0
Solve with the quadratic formula
2u2+2u+1=0
Quadratic Equation Formula:
For a=2,b=2,c=1u1,2​=2⋅2−2±22−4⋅2⋅1​​
u1,2​=2⋅2−2±22−4⋅2⋅1​​
Simplify 22−4⋅2⋅1​:2i
22−4⋅2⋅1​
Multiply the numbers: 4⋅2⋅1=8=22−8​
Apply imaginary number rule: −a​=ia​=i8−22​
−22+8​=2
−22+8​
22=4=−4+8​
Add/Subtract the numbers: −4+8=4=4​
Factor the number: 4=22=22​
Apply radical rule: 22​=2=2
=2i
u1,2​=2⋅2−2±2i​
Separate the solutionsu1​=2⋅2−2+2i​,u2​=2⋅2−2−2i​
u=2⋅2−2+2i​:−21​+i21​
2⋅2−2+2i​
Multiply the numbers: 2⋅2=4=4−2+2i​
Factor −2+2i:2(−1+i)
−2+2i
Rewrite as=−2⋅1+2i
Factor out common term 2=2(−1+i)
=42(−1+i)​
Cancel the common factor: 2=2−1+i​
Rewrite 2−1+i​ in standard complex form: −21​+21​i
2−1+i​
Apply the fraction rule: ca±b​=ca​±cb​2−1+i​=−21​+2i​=−21​+2i​
=−21​+21​i
u=2⋅2−2−2i​:−21​−i21​
2⋅2−2−2i​
Multiply the numbers: 2⋅2=4=4−2−2i​
Factor −2−2i:−2(1+i)
−2−2i
Rewrite as=−2⋅1−2i
Factor out common term 2=−2(1+i)
=−42(1+i)​
Cancel the common factor: 2=−21+i​
Rewrite −21+i​ in standard complex form: −21​−21​i
−21+i​
Apply the fraction rule: ca±b​=ca​±cb​21+i​=−(21​)−(2i​)=−(21​)−(2i​)
Remove parentheses: (a)=a=−21​−2i​
=−21​−21​i
The solutions to the quadratic equation are:u=−21​+i21​,u=−21​−i21​
u=−21​+i21​,u=−21​−i21​
Substitute back u=cot(x)cot(x)=−21​+i21​,cot(x)=−21​−i21​
cot(x)=−21​+i21​,cot(x)=−21​−i21​
cot(x)=−21​+i21​:No Solution
cot(x)=−21​+i21​
NoSolution
cot(x)=−21​−i21​:No Solution
cot(x)=−21​−i21​
NoSolution
Combine all the solutionsNoSolutionforx∈R

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Popular Examples

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Frequently Asked Questions (FAQ)

  • What is the general solution for 2cot(x)+tan(x)=-2 ?

    The general solution for 2cot(x)+tan(x)=-2 is No Solution for x\in\mathbb{R}
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