Proving Trigonometric Equations?

Question

Hello,

There are some questions that I am not sure how to tackle:

1) (1 - tan^2θ) / (1 + tan^2θ) = 1 - 2sin^2θ

2) (2tanθ) / (1 + tan^2θ) = 2sinθ cosθ

3) (cotθ + tanθ) / (cosecθ + secθ) = 1 / (cosθ + sinθ)

4) sin^3x + cos^3x = (sinx + cosx)(1 - sinx cosx)

Thank you in advance.

Answer 1

1) Left hand side identity = 

Apply pythagorean identity

= Right hand side identity.

2) Left hand side identity

Apply pythagorean identity

= Right hand side identity.

Answer 2

3) Left hand side identity

= Right hand side identity

4) Left hand side identity

Apply the formula

Apply pythagorean identity

= Right hand side identity.