Can you tell me

Geometry Level 2

2 sin ⁡ ( x ) − sin ⁡ ( 2 x ) 2 sin ⁡ 3 ( x ) \large \frac {2\sin (x) - \sin (2x)}{2 \sin^3 (x)}

Which of the following option/s is/are equal to the expression above?

\[\begin{array} {} A: & \dfrac 1{2\sin^2 (x)} \\ B: & \dfrac 1{1+\cos (x)} \\ C: & \dfrac 12 \sec^2 \left(\dfrac x2 \right) \end{array} \]

A and B A \text{ and }B A A B B B and C B \text{ and }C C C

Mohamed Sobhy by Mohamed Sobhy , 18, Egypt

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Chew-Seong Cheong
May 17, 2019

2 sin ⁡ x − sin ⁡ 2 x 2 sin ⁡ 3 x = 2 sin ⁡ x − 2 sin ⁡ x cos ⁡ x 2 sin ⁡ 3 x Note that sin ⁡ 2 θ = 2 sin ⁡ θ cos ⁡ θ = 1 − cos ⁡ x sin ⁡ 2 x and sin ⁡ 2 θ + cos ⁡ 2 θ = 1 = 1 − cos ⁡ x 1 − cos ⁡ 2 x = 1 − cos ⁡ x ( 1 − cos ⁡ x ) ( 1 + cos ⁡ x ) = 1 1 + cos ⁡ x . . . ( B : ) also cos ⁡ 2 θ = 2 cos ⁡ 2 θ − 1 = 1 1 + 2 cos ⁡ 2 x 2 − 1 = 1 2 cos ⁡ 2 x 2 = 1 2 sec ⁡ 2 x 2 . . . ( C : ) \begin{aligned} \frac {2 \sin x - \color{#3D99F6}\sin 2x}{2 \sin^3 x} & = \frac {2\sin x - \color{#3D99F6} 2\sin x \cos x}{2\sin^3 x} & \small \color{#3D99F6} \text{Note that }\sin 2\theta = 2 \sin \theta \cos \theta \\ & = \frac {1-\cos x}{\color{#3D99F6} \sin^2 x} & \small \color{#3D99F6} \text{and } \sin^2 \theta + \cos^2 \theta = 1 \\ & = \frac {1-\cos x}{\color{#3D99F6}1-\cos^2 x} \\ & = \frac {1-\cos x}{\color{#3D99F6}(1-\cos x)(1+\cos x)} \\ & = \frac 1{1+\color{#3D99F6}\cos x} \quad ...(B:) & \small \color{#3D99F6} \text{also } \cos 2 \theta = 2\cos^2 \theta - 1 \\ & = \frac 1{1+\color{#3D99F6}2 \cos^2 \frac x2 - 1} \\ & = \frac 1{2\cos^2 \frac x2} \\ & = \frac 12 \sec^2 \frac x2 \quad ...(C:) \end{aligned}

Therefore, the answer is B and C \boxed{B \text{ and }C} .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...