The identity `cos^4 x + 1 - sin^4 x = 2cos^2 x` has to be proved.

`cos^4 x + 1 - sin^4 x`

=> `(1 - sin^2 x)^2 + 1 - sin^4 x`

=> `1 + sin^4 x - 2*sin^2x + 1 - sin^4 x`

=> `2 - 2*sin^2 x`

=> `2(1 - sin^2x)`

=> `2*cos^2x`

**This proves that **`cos^4 x + 1 - sin^4 x = 2cos^2 x `

The identity require to prove :

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

L.H.S -> cos^4 x + 1 - sin^4 x

= cos^4 x - sin^4 x + 1

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

= (cos^2 x + sin^2 x)(c0s^2 x - sin^2 x) + 1 [ using formula : (a - b)^2 = (a)^2 - (b)^2 ]

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

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

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

= cos^2 x + cos^2 x [ since, 1 - sin^2 x = cos^2 x ]

= 2cos^2 x

= R.H.S

**L.H.S =R.H.S Proved **