Prove that: ^224^ - ^26^ = 5-1 8 - Vidyadip

Question

Prove that:
$\sin^224^\circ-\sin^26^\circ=\frac{\sqrt{5}-1}{8}$

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Answer

$\text{LHS}=\sin^224^\circ-\sin^26^\circ$
$=\sin(24+6)\sin(24-6)\\ [\because\sin(\text{A+B})\sin(\text{A}-\text{B}=\sin^2\text{A}-\sin^2\text{B})]$
$=\sin30^\circ\sin18^\circ$
$=\frac{1}{2}.\frac{\sqrt{5}-1}{4}\Big[\because\sin18^\circ=\frac{\sqrt{5}-1}{4}\Big]$
$=\frac{\sqrt{5}-1}{8}$
$=\text{RHS}$

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