Sin^4x+2cos^2x-cos^4=1

Proven. We are done proving this identity because we get a true statement, which is 1 = 1.

Step-by-step explanation:

Sin^4x+2cos^2x-cos^4=1

...

We want to prove : (Sin x) ^ 4 + 2 (cos x) ^2 - (cos x) ^4 = 1

(Sin x) ^ 4 + 2 (cos x) ^2 - (cos x) ^4 = 1

Factor this trinomial. considering ((cos x) ^2) is the variable

so

(((sin x) ^2) ^2 - ((cos x) ^2) ^2 + 2 (cos x) ^2 + 1 - 1 = 1

(((sin x) ^2) ^2 - ((cos x) ^2) ^2 + 2 (cos x) ^2 - 1 + 1 = 1

(((sin x) ^2) ^2 + [ - ((cos x) ^2) ^2 + 2 (cos x) ^2 - 1 ] + 1 = 1

(((sin x) ^2) ^2 - [ ((cos x) ^2) - 1 ]^2 + 1 = 1

But also notice that (sin x) ^2 = 1 - (cos x) ^2 from the trig identity:

(sin x) ^2 + (cos x) ^2 = 1

((1 - (cos x) ^2) ^2 - [ ((cos x) ^2) - 1 ]^2 + 1 = 1

here we see that (1 - (cos x) ^2) ^2 = [ ((cos x) ^2) - 1 ]^2

so we get (0 + 1) = 1

1 = 1 true.

Proven. We are done proving this identity because we get a true statement.