Question

8cos² x - 2cos 2x = 5​

Answer 1

$\huge\underline\bold\green{answer -}$

$\sf{8Cos^{2} x - 2Cos2x = 5}$

$\star$$\sf{\sf\boxed{Cos2x = 2Cos^{2} x - 1}}$

$\sf{8Cos^{2} x - 2(2Cos^{2} x - 1) = 5}$

$\sf{8Cos^{2} x - 4Cos^{2} x + 2 = 5}$

$\sf{4Cos^{2} x = 5 - 2}$

$\sf{4Cos^{2} x = 3}$

$\sf{Cos^{2} x =\dfrac{3}{4}}$

$\sf{Cosx =\sqrt{\dfrac{3}{4}}}$

$\sf{Cosx =\dfrac{\sqrt{3}}{2}}$

$\sf{x = Cos^{-1}(\dfrac{\sqrt{3}}{2})}$

$\sf{x = Cos^{-1}(Cos60)}$

$\sf{\small\boxed{x = 60}}$

=> Extra Explanation of Cos2x :-

$\sf{Cos2x = Cos(x + x)}$

We know that -

$\sf{Cos(x+y)=Cosx Cosy - Sinx Siny}$

So ,

$\sf{Cosx Cosx - Sinx Sinx}$

$\sf{Cos^{2} x - Sin^{2} x}$

$\sf{Cos^{2} x - (1 - Cos^{2} x)}$

$\sf{Cos^{2} x - 1 + Cos^{2} x}$

$\sf{2Cos^{2} x - 1}$