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

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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 `

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vaaruni | High School Teacher | (Level 1) Salutatorian

Posted on

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  

                      

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