Question

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

Answer 1

Answer:

Proven .  We get a true statement of 1 = 1  by transforming the expression on the left side to make it look like the right side.   See below.

Step-by-step explanation:

This  is missing some notation:  Sin^4x+2cos^2x-cos^4=1

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

Replace the  (cos x)^4   with  ((cos x)^2)^2  same with the (sin x)^4 with

((sin x)^2)^2

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

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

Factor the  trinomial   -((cos x)^2)^2  +   2 (cos x)^2   +  1 .  

considering  ((cos x)^2) is the variable

( ((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.