Question

Express $\frac{sin 4\theta}{sin \theta}$ in terms of cos³ θ and cos θ.

Answer 1

we have to express sin4θ/sinθ in term of cos³θ and cosθ

we know, sin2x = 2sinx.cosx

so, sin4θ = 2sin2θ cos2θ

= 2(2sinθ cosθ)cos2θ

= 4sinθ cosθ cos2θ

now, sin4θ/sinθ = 4sinθ cosθ cos2θ/sinθ

= 4cosθ cos2θ

we also know, cos2x = 1 - 2cos²x

so, cos2θ = 1 - 2cos²θ

now, 4cosθ cos2θ = 4cosθ(1 - 2cos²θ)

= 4cosθ - 8cos³θ

hence, sin4θ/sinθ will be 4cosθ - 8cos³θ in term of cos³ θ and cos θ.

Answer 2

HELLO DEAR,




we know:-
sin2x = 2sinx.cosx

then, sin4θ = 2sin2θ cos2θ

=> 2(2sinθ cosθ)cos2θ

=> 4sinθ cosθ cos2θ

now,
sin4θ/sinθ = 4sinθ cosθ cos2θ/sinθ

=> 4cosθ cos2θ

[as, cos2x = 1 - 2cos²x ]

so, cos2θ = 1 - 2cos²θ

now, 4cosθ cos2θ = 4cosθ(1 - 2cos²θ)

= 4cosθ - 8cos³θ



I HOPE IT'S HELP YOU DEAR,
THANKS