I came across a particular equation while using some trigonometric identities, for which I could not derive the other side. The left and right side are equal, but I was unable to derive the right side from the left side.
How do I get to the right side, when given the left side?
In other words: when left=right, should you always be able to rewrite one side to match the other side?
$$ \frac{\sqrt{2+\sqrt{3}}}{4} = \frac{\sqrt{3}+1}{2\sqrt{2}} $$
If you wonder where these values came from (shouldnt matter for the answer), well both sides are equal to: $$\cos{15deg}$$
However, The left side was calculated using the half-angle formula (input the value for cosine of 30 degrees as 2x): $$ \cos{30deg}=\frac{\sqrt{3}}{2}=2x \\ \frac{1+\cos{2x}}{2}=\cos^2{x} $$
The right side was calculated using the angle subtraction formula: $$ \cos{45deg} = \frac{\sqrt{2}}{2} \\ \cos{30deg} = \frac{\sqrt{3}}{2} \\ \cos{45deg-30deg}=\cos{45}*\cos{30}+\sin{45}*\sin{30} $$
The latter is somewhat simpler to solve and gives a nicer result without a root inside a root.