# Verify the identity: `1/(tan^2x) - 1/(cot^2x) = csc^2x - sec^2x`

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### 2 Answers

We have to prove the identity:

`1/(tan^2x) - 1/(cot^2x) = csc^2x - sec^2x`

`1/(tan^2x) - 1/(cot^2x)`

=> `(cos^2x)/(sin^2x) - (sin^2x)/(cos^2x)`

=> `(1 - sin^2x)/(sin^2x) - (1 - cos^2x)/(cos^2x)`

=> `1/(sin^2x) - 1 - 1/(cos^2x) + 1`

=> `1/sin^2x - 1/cos^2x`

=> `csc^2x - sec^2x`

**This proves the identity: **`1/(tan^2x) - 1/(cot^2x) = csc^2x - sec^2x`

L:H:S ≡ 1/tan²x - 1/cot²x

= cot²x - tan²x

**⇒ use the identities cot²θ = cosec²θ - 1 & tan²θ = sec²θ -1 **

= cosec²x - 1 - (sec²x -1)

= cosec²x - sec²x

= R:H:S