Help me prove some Trigonometric Identities?

Question

1. cos^2(1+tan^2)=1
2. sec x tan x(1-sin^2)=sinx
3. csc x(2sin x -(sqrt)2)=0
4. cos ^2(2x)-sin^2(2x)=0
5. sin^2 (theta) csc^2 (theta) = sin^2 (theta) + cos^2 (theta).

Answer 1

1. cos^2x(1+tan^2x)=1

Recall the pythagoras trignometric formulae

sec^2x-tan^2x=1-------------------->1+tan^2x=sec^2x

LHS

=cos^2x(sec^2x)

Recall the  trignometric quotient identity :sec^2x=1/cos^2x

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

=1

Answer 2

2. sec x tan x(1-sin^2x)=sinx

Recall the pythagoras trignometric formulae

sin^2x+cos^2x=1-------------------->1-sin^2x=cos^2x

LHS

=sec x tan x(cos^2x)

Recall the  trignometric quotient identity :secx=1/cosx:tanx=sinx/cosx

=(1/cosx)(sinx/cosx)(cos^2x)

=sinx

Answer 3

5. sin^2 (theta) csc^2 (theta) = sin^2 (theta) + cos^2 (theta).

Recall the pythagoras trignometric formulae

csc^2x-cot^2x=1-------------------->1+cot^2x=csc^2x

LHS

=sin^2 (theta) (1+cot^2(theta))

=sin^2 (theta)+sin^2 (theta)*cot^2(theta)

Recall the  trignometric quotient identity :cot^2x=cos^2x/sin^2x

=sin^2 (theta)+sin^2 (theta)*cos^2(theta)/sin^2 (theta)

=sin^2 (theta)+cos^2(theta)