Question

Prove that : cos^2 x + cos^2(x + 2 pi/3) + cos^2(x − 2 pie/3) = 3/2

Answer 1

Answer:

Step-by-step explanation:

We have,

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

$\tt{=cos^{2}(x)+\left\{cos\left(x\right)\cdot\,cos\left(\dfrac{2\pi}{3}\right)-sin\left(x\right)\cdot\,sin\left(\dfrac{2\pi}{3}\right)\right\}^{2}+}\\\tt{\left\{cos\left(x\right)\cdot\,cos\left(\dfrac{2\pi}{3}\right)+sin\left(x\right)\cdot\,sin\left(\dfrac{2\pi}{3}\right)\right\}^{2}}$

$\rm{\bigstar\,\,\,cos\left(\dfrac{2\pi}{3}\right)=-\dfrac{1}{2}\,\,\,\,\,\&\,\,\,\,\,sin\left(\dfrac{2\pi}{3}\right)=\dfrac{\sqrt{3}}{2}}$

So,

$\tt{=cos^{2}(x)+\left\{-\dfrac{1}{2}\cdot\,cos\left(x\right)-\dfrac{\sqrt{3}}{2}\cdot\,sin\left(x\right)\right\}^{2}+\left\{-\dfrac{1}{2}\cdot\,cos\left(x\right)+\dfrac{\sqrt{3}}{2}\cdot\,sin\left(x\right)\right\}^{2}}$

$\tt{=cos^{2}(x)+\left\{\dfrac{1}{2}\cdot\,cos\left(x\right)+\dfrac{\sqrt{3}}{2}\cdot\,sin\left(x\right)\right\}^{2}+\left\{\dfrac{1}{2}\cdot\,cos\left(x\right)-\dfrac{\sqrt{3}}{2}\cdot\,sin\left(x\right)\right\}^{2}}$

$\tt{=cos^{2}(x)+\left(\dfrac{1}{2}\right)^2\Big\{cos\left(x\right)+\sqrt{3}\,sin\left(x\right)\Big\}^{2}+\left(\dfrac{1}{2}\right)^2\Big\{cos\left(x\right)-\sqrt{3}\,sin\left(x\right)\Big\}^{2}}$

$\tt{=cos^{2}(x)+\dfrac{1}{4}\left[\Big\{cos\left(x\right)+\sqrt{3}\,sin\left(x\right)\Big\}^{2}+\Big\{cos\left(x\right)-\sqrt{3}\,sin\left(x\right)\Big\}^{2}\right]}$

$\bf{\longmapsto\,\,\,\,\left(a+b\right)^2+\left(a-b\right)^2=2\left({a}^{2}+{b}^{2}\right)}$

So,

$\tt{=cos^{2}(x)+\dfrac{1}{4}\cdot2\left[\Big(cos\left(x\right)\Big)^{2}+\Big(\sqrt{3}\,sin\left(x\right)\Big)^{2}\right]}$

$\tt{=cos^{2}(x)+\dfrac{1}{2}\left[cos^{2}\left(x\right)+3\,sin^{2}\left(x\right)\right]}$

$\tt{=cos^{2}(x)+\dfrac{1}{2}\,cos^{2}\left(x\right)+\dfrac{3}{2}\,sin^{2}\left(x\right)}$

$\tt{=\left(1+\dfrac{1}{2}\right)cos^{2}\left(x\right)+\dfrac{3}{2}\,sin^{2}\left(x\right)}$

$\tt{=\left(\dfrac{2+1}{2}\right)cos^{2}\left(x\right)+\dfrac{3}{2}\,sin^{2}\left(x\right)}$

$\tt{=\dfrac{3}{2}\,cos^{2}\left(x\right)+\dfrac{3}{2}\,sin^{2}\left(x\right)}$

$\tt{=\dfrac{3}{2}\Big\{cos^{2}\left(x\right)+sin^{2}\left(x\right)\Big\}}$

$\tt{=\dfrac{3}{2}\cdot1}$

$\tt{=\dfrac{3}{2}}$