Question

Answer the question​

Answer 1

Hence proved: $sin\ 8\theta=8\ sin\theta\ cos\theta\ cos 2\theta\ cos\ 4\theta$.

Step-by-step explanation:

The formula of sin 2θ is,

sin 2θ = 2 sin θ cos θ

Use this formula to compute the value of sin 8θ as follows:

$sin\ 8\theta=sin\ 2(4\theta)$

          $=2sin\ 4\theta\ cos\ 4\theta$

          $=2sin\ 2(2\theta)\ cos\ 4\theta$

          $=2(2\ sin\ 2\theta\ cos 2\theta)\ cos\ 4\theta$

          $=4\ sin\ 2\theta\ cos 2\theta\ cos\ 4\theta$

          $=4\ (2\ sin\theta\ cos \theta)\ cos 2\theta\ cos\ 4\theta$

          $=8\ sin\theta\ cos\theta\ cos 2\theta\ cos\ 4\theta$

Hence proved.

Answer 2

$\displaystyle\longrightarrow\sf{LHS}$

$\displaystyle\longrightarrow\sf{\sin (8\theta)}$

$\displaystyle\longrightarrow\sf{\sin (2\times 4\theta)}$

$\displaystyle\longrightarrow\sf{2\sin (4\theta)\cos (4\theta)}$

$\displaystyle\longrightarrow\sf{2\sin (2\times 2\theta)\cos (4\theta)}$

$\displaystyle\longrightarrow\sf{4\sin (2\theta)\cos (2\theta)\cos (4\theta)}$

$\displaystyle\longrightarrow\sf{8\sin\theta\cos\theta\cos (2\theta)\cos (4\theta)}$

$\displaystyle\longrightarrow\sf{RHS}$