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Prove that:
$2 \sin ^2 \frac{3 \pi}{4}+2 \cos ^2 \frac{\pi}{4}+2 \sec ^2 \frac{\pi}{3}=10$
$2 \sin ^2 \frac{3 \pi}{4}+2 \cos ^2 \frac{\pi}{4}+2 \sec ^2 \frac{\pi}{3}=10$
Solution:
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Verified Answer
LHS $=2 \sin ^2 \frac{3 \pi}{4}+2 \cos ^2 \frac{\pi}{4}+2 \sec ^2 \frac{\pi}{3}$
$=2\left(\sin \frac{3 \pi}{4}\right)^2+2\left(\cos \frac{\pi}{4}\right)^2+2\left(\sec \frac{\pi}{3}\right)^2$
$=2\left(\sin \left[\pi-\frac{\pi}{4}\right]\right)^2+2 \times\left(\frac{1}{\sqrt{2}}\right)^2+2\left(2^2\right)$
$=2\left(\sin \frac{\pi}{4}\right)^2+2 \times \frac{1}{2}+8$
$=2 \times\left(\frac{1}{\sqrt{2}}\right)^2+1+8=2 \times \frac{1}{2}+1+8$
$=1+1+8=10=$ RHS.
$=2\left(\sin \frac{3 \pi}{4}\right)^2+2\left(\cos \frac{\pi}{4}\right)^2+2\left(\sec \frac{\pi}{3}\right)^2$
$=2\left(\sin \left[\pi-\frac{\pi}{4}\right]\right)^2+2 \times\left(\frac{1}{\sqrt{2}}\right)^2+2\left(2^2\right)$
$=2\left(\sin \frac{\pi}{4}\right)^2+2 \times \frac{1}{2}+8$
$=2 \times\left(\frac{1}{\sqrt{2}}\right)^2+1+8=2 \times \frac{1}{2}+1+8$
$=1+1+8=10=$ RHS.
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