Proving Trigonometric Identities Prove: sin^4A + 2cos^2A - cos^4A = 1

Here we have to prove that : (sin A)^4 + 2 (cos A)^2 -
(cos A)^4 = 1.

We use the relation that ( sin A)^2 + (cos
A)^2 = 1

(sin A)^4 + 2 (cos A)^2 - (cos
A)^4

=> [(sin A)^2]^2 + 2 (cos A)^2 - [(cos
A)^2]^2

=> [(sin A)^2]^2 + 2*[1 - (sin A)^2] - [(cos
A)^2]^2

=> [(sin A)^2]^2 + [2 - 2*(sin A)^2] - [(cos
A)^2]^2

=> [(sin A)^2]^2 - [(cos A)^2]^2 + [2 -
2*(sin A)^2]

using a^2 - b^2 = ( a - b)( a +
b)

=> [(sin A)^2 - (cos A)^2][(sin A)^2 + (cos
A)^2]+ [2 - 2*(sin A)^2]

=> [(sin A)^2 - (cos
A)^2]*1 + [2 - 2*(sin A)^2]

=> (sin A)^2 - (cos A)^2
+ 2 - 2*(sin A)^2

=> - (cos A)^2 + 2 - (sin
A)^2

=> - [ (cos A)^2 + (sin A)^2] +
2

=> -1 + 2

=>
1

Therefore (sin A)^4 + 2 (cos A)^2 - (cos
A)^4 = 1.