Class 11MathematicsTrigonometric FunctionsFundamental trigonometrical ratios and functions, Trigonometrical ratio of allied angles
MediumProve that $2 \sin ^{2} \frac{3 \pi}{4} + 2 \cos ^{2} \frac{\pi}{4} + 2 \sec ^{2} \frac{\pi}{3} = 10$.
(N/A) $L.H.S. = 2 \sin ^{2} \frac{3 \pi}{4} + 2 \cos ^{2} \frac{\pi}{4} + 2 \sec ^{2} \frac{\pi}{3}$
$= 2 \left\{ \sin \left( \pi - \frac{\pi}{4} \right) \right\}^{2} + 2 \left( \frac{1}{\sqrt{2}} \right)^{2} + 2 (2)^{2}$
$= 2 \left( \sin \frac{\pi}{4} \right)^{2} + 2 \times \frac{1}{2} + 2(4)$
$= 2 \left( \frac{1}{\sqrt{2}} \right)^{2} + 1 + 8$
$= 2 \left( \frac{1}{2} \right) + 1 + 8$
$= 1 + 1 + 8 = 10$
$= R.H.S.$
$= 2 \left\{ \sin \left( \pi - \frac{\pi}{4} \right) \right\}^{2} + 2 \left( \frac{1}{\sqrt{2}} \right)^{2} + 2 (2)^{2}$
$= 2 \left( \sin \frac{\pi}{4} \right)^{2} + 2 \times \frac{1}{2} + 2(4)$
$= 2 \left( \frac{1}{\sqrt{2}} \right)^{2} + 1 + 8$
$= 2 \left( \frac{1}{2} \right) + 1 + 8$
$= 1 + 1 + 8 = 10$
$= R.H.S.$
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