Prove that (tanx)^2-(sinx)^2=(tanx)^2(sinx)^2

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justaguide eNotes educator| Certified Educator

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

Let's start from the left hand side:

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

tan x = sin x / cos x

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

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

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

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

This is the right hand side.

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

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