Question

Solve the following equations.
$sinxsin7x=sin3xsin5x.$

Answer 1

$sinx.sin7x=sin3x.sin5x$

we know that,

$\left\{cos\right(A-B)-cos(A+B)=2sinA.sinB\}$

$\Rightarrow 2sinx.sin7x=2sin3x.sin5x$

$\Rightarrow cos6x-cos8x=cos2x-cos8x$

$\Rightarrow cos6x-cos2x=0$

also we know that,

$\{cosC-cosD=2sin\left(\frac{C+D}{2}\right).sin\left(\frac{D-C}{2}\right)\}$

$\Rightarrow 2sin\left(\frac{6x+2x}{2}\right).sin\left(\frac{2x-6x}{2}\right)=0$

$\Rightarrow 2sin4x.sin(-2x)=0$

$\Rightarrow -2sin4x.sin2x=0...\{\because sin(-\theta )=-sin\theta \}$

$\Rightarrow sin4x=0$ or $sin2x=0$

we know , the general solution when

$sin\theta =0\Rightarrow \{\theta =n\pi ,nϵZ\}=Z\to$ integers $(0,\pm 1,\pm 2,\pm 3\pm ...)$

for $\theta =4x$

$sin4x=0\Rightarrow \{4x=n\pi ,nϵZ\}$

$\Rightarrow \{x=n\frac{\pi }{4},nϵZ\}$

also for $\theta =2x$

$sinx=0\Rightarrow \{2x=n\pi ,nϵZ\}$

$\Rightarrow \{x=n\frac{\pi }{2},nϵZ\}$