Question

Prove that 2sin 3pi/4 + 2cos pi/4 + 2sec pi/3 =10

Answer 1

Correct Question :-

To Prove :-

$\sf 2sin^2 \dfrac{ 3 \pi}{4} + 2cos^2 \dfrac{\pi}{4} +2sec^2 \dfrac{\pi}{3} = 10$

Solution :-

$\bullet \sf \ sin \dfrac{3\pi}{4} = sin \left( \pi - \dfrac{\pi}{4} \right)$

            $= \sf sin \dfrac{\pi}{4} \\\\= \dfrac{1}{\sqrt{2}}$

$\bullet \sf\ cos \dfrac{\pi}{4} = \dfrac{1}{\sqrt{2}}$

$\bullet \sf \ sec \dfrac{\pi}{3} = \dfrac{1}{cos \dfrac{\pi}{3}}$

           $= \dfrac{1}{\dfrac{1}{2}} \\\\= 2$

$\longrightarrow \sf 2sin^2 \dfrac{3\pi}{4}+2cos^2 \dfrac{\pi}{4} +2sec^2 \dfrac{\pi}{3} \\\\\longrightarrow 2 \times \left( \dfrac{1}{\sqrt{2}} \right)^2+ 2 \times \left( \dfrac{1}{\sqrt{2}} \right)^2 + 2 \times 2^2 \\\\\longrightarrow \dfrac{2}{2} +\dfrac{2}{2} + 2 \times 4 \\\\\longrightarrow 1+1+8 \\\\\longrightarrow 10 \\\\\longrightarrow R.H.S$

Hence proved.

Answer 2

$\huge\underline{\bf \orange{AηsωeR :}}$

3π/4 =sin(π − π/4)

= sin π/4 = 1/2

π/4 = 1/2cos

π/3 = 1cos π/3

= 1 ½=2

⟶2sin²3π/4 +2cos² π/4 +2sec² π/3

⟶ n2×( ½)²+2×( ½ )² +2×2²

⟶ 2/2 + 2/2 +2×4

⟶ 1+1+8

⟶ 10

⟶ R.H.S

Hence proved.