Question

solve : 4sinx.sin2x.sin4x=sin3x

Answer 1

solution::x=nπ,. x=2nπ+-π÷9

Answer 2

Answer:

The value of the given expression is x=nπ and x= mπ/3 ± π/9 .

Step-by-step explanation:

Using two formulas

$[\sin (A-B)][\sin (A+B)]=\sin ^{2} A-\sin ^{2} B \text { and } \sin 3 x=3 \sin x-4 \sin 3 x$

On applying above formula and simplifying the equation 4sinx.sin2x.sin4x=sin3x, we get

4sinx x sin(3x-x) sin(3x+x)= sin 3x

Simplifying further, we get

$\begin{array}{l}{4 \sin x\left(\sin ^{2} 3 x-\sin ^{2} x\right)=3 \sin x-4 \sin ^{3} x} \\ {4 \sin x\left(\sin ^{2} 3 x-4 \sin ^{3} x\right)=3 \sin x-4 \sin ^{3} x}\end{array}$

$\begin{array}{l}{4 \sin x \sin ^{2} 3 x-2 \sin x=0} \\ {\sin \left(4 \sin ^{2} 3 x-3\right)=0}\end{array}$

Splitting the above terms into two groups, we get two sin terms

Term 1: sinx=0  

x=nπ

Term 2: $4 \sin ^{2} 3 x-3=0$

$\sin ^{2} 3 x=3 / 4=\sqrt{\frac{3}{2}^{2}}$

$=\sin ^{2}(\pi / 3)$ where π is an angle in trigonometry

3x=mπ ± π/3

x= mπ/3 ± π/9

Where n and m are integers