Question

Prove that, 2sin^2 3π/4 +2cos^2 π/4 + 2sec^2 π/3 = 10 ​

Answer 1

$\bold{\huge{\underline{Question}}}$

$\bf \: Prove \: that \: {2 \: sin}^{2} \frac{3\pi}{4} + {2 \: cos}^{2} \frac{\pi}{4} + {2 \: sec}^{2} \frac{\pi}{3} = 10$

$\bold{\huge{\underline{Solution-}}}$

$\bf \: Used \: formulas \\ \: \: \: \: \: \: \boxed{ \bf \: 1. \: \: {sin}^{2}x \: + {cos}^{2}x \: = 1 } \\ \: \: \: \: \: \:\boxed{ \bf \: 2. \: \pi = 180 \degree }$

$\underline{ \large \bf \: L.H.S} =$

$\bf \: {2 \: sin}^{2} \frac{3\pi}{4} + {2 \: cos}^{2} \frac{\pi}{4} + 2 \: {sec}^{2} \frac{\pi}{3} \\ \\ \implies \bf \: 2 \{ \: sin {}^{2} \frac{3\pi}{4} + {cos}^{2} \frac{\pi}{4} + {sec}^{2} \frac{\pi}{3} \} \\ \\ \implies \bf \: 2 \{ \: {sin}^{2} (\pi - \frac{\pi}{4} ) + {cos}^{2} \frac{\pi}{4} + {sec}^{2} \frac{\pi}{3} \\ \\ \implies \bf \: 2 \{( {sin}^{2} \frac{\pi}{4} + {cos}^{2} \frac{\pi}{4} ) + {sec}^{2} \frac{\pi}{3} \} \\ \\ \implies \bf \: 2 \{1 + (sec \frac{180 \degree}{3} ) {}^{2} \} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \huge \{ \tiny \: sin {}^{2} x + {cos}^{2} x = 1 \\ \\ \implies \bf \: 2 \{ \: 1 + (sec \: 60 \degree) {}^{2} \} \: \\ \\ \implies \bf \: 2 \{ \: 1 + {2}^{2} \} \\ \\ \implies \bf \: 2 \{1 + 4 \} \\ \\ \implies \bf \: 10$

Answer 2

2sin^2 3π/4 +2cos^2 π/4 + 2sec^2 π/3

=2 sin^2 3*180/4 +2 cos^2 180/4 +2 sec^2 2*180/3

= 2 sin ^2 3*45+2 cos^2 45 + 2 sec^ 120

= 2 sin^ 135 + 2{ 1/$\sqrt{2}$}^2 +2 (- sec 60 )^2

= 2 (cos 45)^2+2*1/2+ 2 *2^2

=2 *{$1/\sqrt{2}$}^2 +1+8

= 2*1/2+1+8

= 1+9

= 10