Question

$\mathbb{PROVE QUESTION} \\ \\ \longrightarrow \rm \cos^{2} x + \cos ^{2} \bigg\{ x + \dfrac{\pi}{3} \bigg\} + \cos^{2} \bigg\{ x - \dfrac{\pi}{3} = 3/2. \mathbb{NOTE IRRELEVANT ANSWER WILL BE REPORTED}​$

Answer 1

Answer

3/2

Formula Used

$\rm cos2\theta+1=2cos^2\theta\\\\\bullet \;\; \rm cos^2\theta=\dfrac{cos2\theta+1}{2}\\\\\bullet \;\; \rm{\tiny{cos(A)+cos(B)=2cos\left(\dfrac{A+B}{2}\right).cos\left(\dfrac{A-B}{2}\right)}}\\\\\bullet \;\; \rm cos(\pi-\theta)=-sin\theta$

Solution

LHS

$\rm \to cos^2x+cos^2\left(x+\dfrac{\pi}{3}\right)+cos^2\left(x-\dfrac{\pi}{3}\right)$

$\rm\to \tiny{\dfrac{cos2x+1}{2}+\dfrac{cos2\left(x+\dfrac{\pi}{3}\right)+1}{2}+\dfrac{cos2\left(x-\dfrac{\pi}{3}\right)+1}{2}}\\\\\to \rm \tiny{\dfrac{1}{2}\left( 3+cos2x+cos\left(2x+\dfrac{2\pi}{3}\right)+cos\left(2x-\dfrac{2\pi}{3}\right)\right)}\\\\\to \rm \tiny{\dfrac{1}{2}\rm \left(3+cos2x+2cos\left(\dfrac{2x+\dfrac{2\pi}{3}+2x-\dfrac{2\pi}{3}}{2}\right).cos\left(\dfrac{2x+\dfrac{2\pi}{3}-2x+\dfrac{2\pi}{3}}{2}\right)\right) }$

$\rm \to \dfrac{1}{2}\left(3+cos2x+2cos2x.(-sin30^0)\right)\\\\\rm \to \dfrac{1}{2}\left(3+cos2x+2cos2x.\dfrac{-1}{2}\right)\\\\\rm \to \dfrac{1}{2}\left(3+cos2x-cos2x\right)\\\\\rm \to \dfrac{3}{2}\\\\\rm \to RHS$

Hence proved

Answer 2

Answer:

3/2

Step-by-step explanation:

LSH. ${cos}^{2} + {cos }^{2}(x + \frac{\pi}{3} ) + {cos}^{2}(x - \frac{\pi}{3})$