Question

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

Accepted Answer

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

Answer

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

Math

Prove: `cos^4 x + 1 - sin^4 x = 2cos^2 x`

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