Question

sinsin6x- sinsin4x=sin2x sin10x
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Answer 1

$\bold{LHS =sin^26x−sin^24x}$

=(sin6x+sin4x)(sin6x−sin4x)

$\bold{... [∵a^2 −b^2 =(a+b)(a−b)]}$

We know that, sinA+sinB=

$2 \sin( \frac{A+B}{2} ) \cos( \frac{A - B}{2} )$

∴ LHS =

$[2 \sin( \frac{6x + 4x}{2} ) \cos( \frac{6x - 4x}{2} ) ]$

$[2 \cos( \frac{6x + 4x}{2} ) \sin( \frac{6x - 4x}{2} ) ]$

$= [2 \sin( \frac{10x}{2} ) \cos( \frac{2x}{2} ) ]$

$= [2 \cos( \frac{10x}{2} ) \sin( \frac{2x}{2} ) ]$

=2sin5xcosx×2cos5xsinx

=2sin5xcos5x×2sinxcosx

=sin10x×sin2x [∵sin2θ=2sinθcosθ]

= RHS

Hence proved.