Find a solution to the equation\frac{1}{\sin^2 (2x)}+\tan(x)-\frac{1}{\tan(x)}=2Assuming \sin(x) \neq 0

Haialarmz6

Haialarmz6

Answered question

2022-01-22

Find a solution to the equation
1sin2(2x)+tan(x)1tan(x)=2
Assuming sin(x)0 and cos(x)0, I simplified the above to
4sin2(x)14sin(x)cos(x)=0
which is equivalent to
cos(2x)+sin(2x)=12
I do not know what to do next. I would be grateful for any hints.

Answer & Explanation

Jordyn Horne

Jordyn Horne

Beginner2022-01-23Added 16 answers

We have that by linear combination of sine and cosine
cos(2x)+sin(2x)=2sin(2x+π4)=12
immablondevl

immablondevl

Beginner2022-01-24Added 11 answers

Use tan2A=2tanA1tan2A
1tanxtanx==2cot2x
1sin22x=1+cot22x
On simplification
1=2tan2x1tan22x=tan4x

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