Check the solution to "Proof trigonometric identities. tan4(x)−1sec2(x)(tan⁡(x)−1)=tan⁡(x)+1"

Isa Trevino

Answered question

2021-09-08

Proof trigonometric identities.

\tan^{4} (x) - \frac{1}{\sec^{2} (x) (\tan (x) - 1)} = \tan (x) + 1

unessodopunsep

Skilled2021-09-09Added 105 answers

Given that

\frac{\tan^{4} (x) - 1}{\sec^{2} (x) (\tan (x) - 1)} = \tan (x) + 1

To prove the given identity
Let LHS

\frac{\tan^{4} (x) - 1}{\sec^{2} (x) (\tan (x) - 1)}

\Rightarrow \frac{{(\tan^{2} x)}^{2} - {(1)}^{2}}{\sec^{2} x (\tan x - 1)}

\Rightarrow \frac{(\tan^{2} x + 1) (\tan^{2} x - 1)}{\sec^{2} x (\tan x - 1)}

\Rightarrow \frac{(\tan^{2} x + 1) (\tan x + 1) (\tan x - 1)}{\sec^{2} x (\tan x - 1)}

\Rightarrow \frac{(\tan^{2} x + 1) (\tan x + 1)}{\sec^{2} x}

\Rightarrow \frac{(\tan^{2} x + 1) (\tan x + 1)}{\tan^{2} x + 1}

\Rightarrow \tan x + 1

Hence from equation L.H.S.=R.H.S

Jeffrey Jordon

Expert2022-01-31Added 2605 answers

Answer is given below (on video)

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