Prove: tan^2x - sin^2x = tan^2x sin^2x

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We have to prove that (tan x)^2 - (sin x)^2 = (tan x)^2 * (sin x)^2

Start from the left hand side:

(tan x)^2 - (sin x)^2

use tan x = sin x / cos x

(sin x)^2/(cos x)^2 - (sin x)^2

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

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We have to prove that (tan x)^2 - (sin x)^2 = (tan x)^2 * (sin x)^2

Start from the left hand side:

(tan x)^2 - (sin x)^2

use tan x = sin x / cos x

(sin x)^2/(cos x)^2 - (sin x)^2

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

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

=> (sin x)^2 * [(sin x)^2 / (cos x)^2]

=> (tan x)^2 * (sin x)^2

which is the right hand side.

This proves that (tan x)^2 - (sin x)^2 = (tan x)^2 * (sin x)^2

Approved by eNotes Editorial Team