Prove that: 4( ^310^ + ^320^ )=3( 10^ + 20^ ) - Vidyadip

Question

Prove that:
$4(\cos^310^\circ+\sin^320^\circ)=3(\cos10^\circ+\sin20^\circ)$

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Answer

Consider the LHS of the given equation
$4(\cos^310^\circ+\sin^320^\circ)=3(\cos10^\circ+\sin20^\circ)$
$\text{LHS} = 4(\cos^310^\circ+\sin^320^\circ)$
since $\sin30=\cos60^\circ=\frac{1}{2}$
and $\sin60^\circ=\cos30^\circ=\frac{\sqrt{3}}{2}$
$\Rightarrow\sin3.20^\circ=\cos3.10^\circ$
$\Rightarrow3\sin20^\circ-4\sin^320^\circ=4\cos^310^\circ-3\cos10^\circ$
$\Rightarrow4(\cos^310^\circ+\sin^320^\circ)=3(\cos10^\circ+\sin20^\circ)$
$=\text{RHS}$

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