Prove that:4 ( ^4 30^ + ^4 60^ ) -3 ( ^2 45^ - ^2 90^ ) = 2 - Vidyadip

Question

Prove that:$4 (\sin^4 30^\circ + \cos^4 60^\circ ) -3 (\cos^2 45^\circ - \sin^2 90^\circ ) = 2$

✓

Answer

$\text{LHS} =4\left(\sin ^4 30^{\circ}+\cos ^4 60^{\circ}\right)-3\left(\cos ^2 45^{\circ}-\sin ^2 90^{\circ}\right)$
$=4\left[\left(\frac{1}{2}\right)^4+\left(\frac{1}{2}\right)^4\right]-3\left[\left(\frac{1}{\sqrt{2}}\right)^2+(1)^4\right]$
$=4\left[\frac{1}{16}+\frac{1}{16}\right]-3\left[\frac{1}{2}-1\right]=\frac{4 \times 2}{16}+3 \times \frac{1}{2}=2$
$\text{RHS} = 2$
$\text{LHS = RHS}$

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