Verify each trigonometric equation by substituting identities

Question

1 .cot x sec4x = cot x + 2 tan x + tan3x

2. (sin x)(tan x cos x - cot x cos x) = 1 - 2 cos2x

3. 1 + sec2x sin2x = sec2x

4. - tan2x + sec2x = 1 Please help me out and I just need answers. No explanations please.

Answer 1

1) To prove

                   

Multiply numerator and denominator with cosx

Hence proved

2)To prove

Hence proved

3) To prove

Hence proved

4) To prove

We have an identity that

Therefore

Hence proved

Answer 2

4).

The equation is - tan² x + sec² x = 1.

Left hand side identity : - tan² x + sec² x.

Trigonometric function : tan x = sin x / cos x,

Reciprocal identity : sec x = 1/cos x.

- tan² x + sec² x = - (sin x/cos x)² + (1/cos x)²

= - sin² x/cos² x + 1/cos² x

= (1 - sin² x)/cos² x

Pythagorean identity : sin² x + cos² x = 1 .

= cos² x / cos² x

= 1

= Right hand side identity.

Hence proved.