sin² 6x - sin² 4x = sin 2x sin 10x
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LHS =sin
2
6x−sin
2
4x
=(sin6x+sin4x)(sin6x−sin4x) ... [∵a
2
−b
2
=(a+b)(a−b)]
We know that, sinA+sinB=2sin(
2
A+B
)cos(
2
A−B
)
∴ LHS =[2sin(
2
6x+4x
)cos(
2
6x−4x
)][2cos(
2
6x+4x
)sin(
2
6x−4x
)]
=[2sin(
2
10x
)cos(
2
2x
)][2cos(
2
10x
)sin(
2
2x
)]
=2sin5xcosx×2cos5xsinx
=2sin5xcos5x×2sinxcosx
=sin10x×sin2x ...[∵sin2θ=2sinθcosθ]
= RHS
Hence proved.
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