Q. 1

# Prove that:sin 5x = 5 sin x – 20 sin3 x + 16 sin5 x

LHS is

sin 5x = sin(3x+2x)

But we know,

sin(x+y) = sin x cos y+cos x sin y…..(i)

sin 5x = sin 3x cos 2x+cos 3x sin 2x

sin 5x = sin (2x+x) cos 2x+cos (2x+x) sin 2x……..(ii)

And

cos (x+y) = cos(x)cos(y) – sin(x)sin(y)……(iii)

Now substituting equation (i) and (iii) in equation (ii), we get

sin 5x = (sin 2x cos x+cos 2x sin x )cos 2x+( cos 2x cos x – sin 2x sin x) sin 2x

sin 5x = sin 2x cos 2x cos x+cos2 2x sin x+(sin 2x cos 2x cos x – sin2 2x sin x)

sin 5x = 2sin 2x cos 2x cos x+cos2 2x sin x– sin2 2x sin x …….(iv)

Now sin 2x = 2sin x cos x………(v)

And cos 2x = cos2x –sin2x………(vi)

Substituting equation (v) and (vi) in equation (iv), we get

sin 5x = 2(2sin x cos x)(cos2x –sin2x)cos x+(cos2x –sin2x)2sin x –(2sin x cos x)2sin x

sin 5x = 4(sin x cos2 x)([1–sin2x] –sin2x)+([1–sin2x]–sin2x)2sin x –(4sin2 x cos2 x)sin x (as cos2x+sin2x=1 cos2x=1–sin2x)

sin 5x = 4(sin x [1–sin2x])(1–2sin2x)+(1–2sin2x)2sin x–4sin3 x [1–sin2x]

sin 5x = 4sin x(1–sin2x)(1–2sin2x)+(1–4sin2x+4sin4x)sin x–4sin3 x +4sin5x

sin 5x = (4sin x–4sin3x)( 1–2sin2x) +sin x–4sin3x+4sin5x–4sin3 x +4sin5x

sin 5x = 4sin x–8sin3x–4sin3x+8sin5x+sin x–8sin3x+8sin5x

sin 5x = 5sin x–20sin3x+16sin5x

Hence LHS = RHS

[Hence proved]

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