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