Help on Trigonometric Identities?

Question

csc^4 x – cot^4 x = csc^2 x + cot^2 x

(cosx + 1)/(sin^3 x)=(csc x)/(1 – cosx)

(sinθ + cosθ )^2+ (sinθ - cosθ )^2= 2

Answer 1

1) Left hand side identity = csc⁴x - cot⁴x

= (csc²x)² - (cot²x)²

{ From pythagorean identities cot²x + 1 = csc²x }

= (cot²x + 1)² - (cot²x)²

{ Apply the formula (a + b)²  = a² + b² + 2ab, In this case a = cotx , b = 1}

= cot⁴x + 1 + 2cot²x - cot⁴x

= 1 + 2cot²x

= 1 + cot²x + cot²x

= (1 + cot²x) + cot²x

= csc²x + cot²x

= Right hand side identity.

Answer 2

2) Left hand side identity

From pythagorean identities  sin²(x) + cos²(x) = 1

= Right hand side identity.

Answer 3

3) Left hand side identity = ( sinθ + cosθ )² + ( sinθ - cosθ )²

{ Apply the formula (a + b)²  = a² + b² + 2ab }

= [ sin²θ + cos²θ + 2 sinθ cosθ ]+ [ sin²θ + cos²θ - 2 sinθ cosθ]

From pythagorean identities  sin²(θ) + cos²(θ) = 1

= [1 + 2 sinθ  cosθ] + [ 1 - 2 sinθ  cosθ]

= 1 + 2 sinθ  cosθ +  1 - 2 sinθ  cosθ

= 2

Right hand side identity.