Find the value of m

Question

I was doing some trig problems for leisure. This one particularly seems not trivial. So I thought someone may be interested to take a look.

Answer 1

If we assume the claim is true and substitute $x = 0$, we find $4-m=0$, which is straightforward to solve for $m$.

If we'd like to check another angle, try $x = \pi/4$ and discover $2 = \frac{m}{2}$.

Finding $m$ is fairly easy: pick a value of $x$ and solve the resulting linear equation for $m$. If we want a little work, take $x = \pi/3$ and obtain $\frac{5}{2} + \frac{m}{2} = \frac{9m}{8}$.

Answer 2

By using following identities, $$\cos(2x)=2\cos^2(x) - 1$$ and $$\sin^2(x)=\dfrac{1-\cos(2x)}{2}$$ we can simplify the given expression as, $$2\cos^2 x-1-m \cos(2x)+3=2 m \left({\dfrac{1-\cos(2x)}{2}}\right)^2$$ $$\Rightarrow [4-m][\cos^2(2x)+1]=0$$ $$\Rightarrow m=4$$ and $x$ belongs to real.