Question

How to prove this identity? `sin^2x+tan^2x * sin^2x= tan^2x`

Answer 1

Shown below...

Explanation:

Use our trig identities...

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

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

`=> tan^2 x + 1 = 1/cos^2 x`

Factor the left side of your problem...

`=> sin^2 x ( 1 + tan^2 x )`

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

`=> (sinx/cosx)^2 = tan^2 x`

Answer 2

Given,

`sin^2 x +tan^2x sin^2x`

`=sin^2 x(1+tan^2 x)`

`=sin ^2 x sec ^2x` (as,`sec^2x - tan^2 x=1)`

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

`=tan^2 x`

              Proved