Question

How do you prove `\frac{ 1+ \tan ^ { 2} x }{ \sin ^ { 2} x + \cos ^ { 2} x } = \sec ^ { 2} x`?

Answer 1

First of all, apply the identity `sin^2x + cos^2x = 1` to get rid of the denominator.

`(1 + tan^2x)/(sin^2x + cos^2x) = sec^2x`

`(1 + tan^2x)/1 = sec^2x`

`1 + tan^2x = sec^2x`

Use the identities `tanx = sinx/cosx` and `secx = 1/cosx`.

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

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

Apply the pythagorean identity mentioned in the first line.

`1/cos^2x = 1/cos^2x`

`LHS = RHS`

Hopefully this helps!