Answer:
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.
