if cosx-sinx=p and cos3x+sin3x=q,prove that q=3p-2p^3

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The expression cos x - sin x = p and cos 3x + sin 3x = q.

3p - 2p^3

= `3*(cos x - sin x) - 2*(cos x - sin x)^3`

= `3*cos x - 3*sin x - 2*(cos^2 x + sin^2 x - 2*sin x*cos x)(cos x - sin x)`

= `3*cos x - 3*sin x - 2*(1 - 2*sin x*cos x)(cos x - sin x)`

= `3*cos x - 3*sin x + 2*(1 - 2*sin x*cos x)(sin x - cos x)`

= `3*cos x - 3*sin x + 2*sin x - 2*cos x - 4*sin^2x*cos x + 4*sin x*cos^2 x `

= `cos x - sin x - 4*sin^2x*cos x + 4*sin x*cos^2 x`

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

= `cos x - sin x - 4*cos x + 4*cos^3 x + 4*sin x - 4*sin^3 x`

= `3*sin x - 3*cos x + 4*cos^3 x - 4*sin^3 x` ...(1)

q = cos 3x + sin 3x

= `3*sin x - 4*sin^3 x + 4cos^3x - 3 cos x` ...(2)

(1) = (2)

This proves that q = 3p - 2p^3

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