Prove `(cot^2 x -1)*sin^2 x = cos 2x`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the trigonometric formula `cot^2 x + 1 = 1/(sin^2 x).` Replacing `cot^2 x`  by `1/ sin^2 x - 1`  yields:

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

Bringing the terms in the brackets to a common denominator yields:

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

`` Reducing by `sin^2 x`  yields:

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

The last equation proves that the identity `(cot^2 x- 1)*sin^2 x = cos 2x`  is checked.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to prove: `(cot^2x - 1)*sin^2x = cos 2x`

Start from the left hand side: `(cot^2x - 1)(sin^2x)`

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

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

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

=> cos 2x

which is the right hand side

This proves: `(cot^2x - 1)(sin^2x) = cos 2x`

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