Question -
Answer -
Let us consider LHS:
sin 5x
Now,
sin 5x = sin (3x + 2x)
But we know,
Sin (x + y) = sin x cos y + cos x sin y…..(i)
So,
sin 5x = sin 3x cos 2x + cos 3x sin 2x
= 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 + ( cos2x cos x – sin 2x sin x) sin 2x
= sin 2x cos 2x cos x + cos2 2x sin x+ (sin 2x cos 2x cos x – sin2 2x sin x)
= 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)2 sin x – (2sin xcos x)2 sin x
= 4(sin x cos2 x) ([1– sin2x]– sin2x) + ([1–sin2x] – sin2x)2 sinx – (4sin2 x cos2 x)sin x
(as cos2x + sin2x = 1 ⇒ cos2x =1– sin2x)
sin 5x = 4(sin x [1 – sin2x]) (1 – 2sin2x)+ (1 – 2sin2x)2 sin x – 4sin3 x [1 –sin2x]
= 4sin x (1 – sin2x) (1 – 2sin2 x)+ (1 – 4sin2x + 4sin4x) sin x – 4sin3 x +4sin5x
= (4sin x – 4sin3x) (1 – 2sin2x)+ sin x – 4sin3x + 4sin5x – 4sin3 x +4sin5x
= 4sin x – 8sin3x – 4sin3x +8sin5x + sin x – 8sin3x + 8sin5x
= 5sin x – 20sin3x + 16sin5x
= RHS
Hence proved.