8. sin 5x =5 cos ^4x sinx - 10 cos^2 x sin^3x+ sin^5x
prove that
Answers
Answer:
Sin5x = Sin⁵x + 5Cos⁴xSinx + 6Sin³xCos²x
Step-by-step explanation:
sin 5x =5 cos ^4x sinx - 10 cos^2 x sin^3x+ sin^5x
prove that
Sin 5x = Sin(3x + 2x)
using Sin(A + B) = SinaCosB + CosACosB
= Sin3xCos2x + Cos3xSin2x
using Sin3A = 3SinA - 4Sin³A
Sin2A = 2SinACosA
Cos3A = 4Cos³A - 3CosA
Cos2A = 2Cos²A - 1
= (3Sinx - 4Sin³x)(2Cos²x - 1) + (4Cos³x - 3Cosx)(2SinxCosx)
= 6SinxCos²x - 8Sin³xCos²x -3Sinx + 4Sin³x + 8Cos⁴xSinx - 6Cos²xSinx
= -8Sin³xCos²x -3Sinx + 4Sin³x + 8Cos⁴xSinx
= -Sin³xCos²x - 7Sin³xCos²x -3Sinx + 4Sin³x + 3Cos⁴xSinx + 5Cos⁴xSinx
= -Sin³x(1 - Sin²x) - 7Sin³xCos²x -3Sinx + 4Sin³x + 3Cos²x(cos²x)Sinx + 5Cos⁴xSinx
= Sin⁵x - Sin³x - 7Sin³xCos²x -3Sinx + 4Sin³x + 3Cos²x(1 - Sin²x)Sinx + 5Cos⁴xSinx
= Sin⁵x + 5Cos⁴xSinx - 7Sin³xCos²x -3Sinx + 3Sin³x + 3Cos²x(1 - Sin²x)Sinx
= Sin⁵x + 5Cos⁴xSinx - 7Sin³xCos²x - 3Sinx + 3Sin³x + 3Cos²xSinx - 3Cos²xSin³x
= Sin⁵x + 5Cos⁴xSinx - 10Sin³xCos²x - 3Sinx + 3Sin³x + 3(1 - Sin²x)Sinx
= Sin⁵x + 5Cos⁴xSinx - 10Sin³xCos²x - 3Sinx + 3Sin³x + 3(1 - Sin²x)Sinx
= Sin⁵x + 5Cos⁴xSinx + 6Sin³xCos²x + 0
= Sin⁵x + 5Cos⁴xSinx + 6Sin³xCos²x
= RHS
= Proved
Sin5x = Sin⁵x + 5Cos⁴xSinx + 6Sin³xCos²x