Given $2\arcsin(x)-3\arccos(x)=\frac{\pi}6 $, find the value of x.

Question

I know that $\arcsin(x) + \arccos(x) = \frac{\pi}2$,

but how to use that to solve the following question?

$$2\arcsin(x)-3\arccos(x)=\frac{\pi}6 $$

Answer 1

Hint:

Rearrange your known identity into: $$\arcsin(x) =\frac{\pi}2-\arccos(x)$$

and substitute it into the equation. Then the equation should be in terms of $\arccos x$. Solve for $x$.

Answer 2

Eliminate either arcsin or arcos.

If arcsin is eliminated, using both equations and simplifying $ x= \cos (\pi/6)= \sqrt3 /2. $

(The first one is an identity. The second one is an equation to be solved for $x$. An identity can be used to simplify part of the equation but not solve it using only this identity).