# How do you simplify 4sinx + 5sin6x ?

Jun 17, 2018

$\text{The Exp.} = 2 \sin x \left\{2 + 5 \cos x \left(3 - 16 {\sin}^{2} x + 16 {\sin}^{4} x\right)\right\}$.

#### Explanation:

$\text{The Expression} = 4 \sin x + 5 \sin 6 x$,

$= 4 \sin x + 5 \left(2 \textcolor{red}{\sin 3 x} \textcolor{b l u e}{\cos 3 x}\right)$,

$= 4 \sin x + 10 \textcolor{red}{\left(3 \sin x - 4 {\sin}^{3} x\right)} \textcolor{b l u e}{\left(4 {\cos}^{3} x - 3 \cos x\right)}$,

$= 4 \sin x + 10 \left\{\sin x \left(3 - 4 {\sin}^{2} x\right)\right\} \left\{\cos x \left(4 {\cos}^{2} x - 3\right)\right\}$,

$= 4 \sin x + 10 \sin x \cos x \left(3 - 4 {\sin}^{2} x\right) \left(4 {\cos}^{2} x - 3\right)$,

$= 4 \sin x + 10 \sin x \cos x \left(3 - 4 {\sin}^{2} x\right) \left\{4 \left(1 - {\sin}^{2} x\right) - 3\right\}$,

$= 4 \sin x + 10 \sin x \cos x \left(3 - 4 {\sin}^{2} x\right) \left(1 - 4 {\sin}^{2} x\right)$,

$= 4 \sin x + 10 \sin x \cos x \left(3 - 16 {\sin}^{2} x + 16 {\sin}^{4} x\right)$.

$\Rightarrow \text{The Exp.} = 2 \sin x \left\{2 + 5 \cos x \left(3 - 16 {\sin}^{2} x + 16 {\sin}^{4} x\right)\right\}$.