Evaluate and Simplify $\cos[\cos^{-1}(\frac{3}{5}) + \frac{\pi}{3}]$ I am trying to evaluate and simplify $\cos[\cos^{-1}(\frac{3}{5}) + \frac{\pi}{3}]$.

I am getting $\frac{11}{10}$ but the answer is $\frac{3-4\sqrt{3}}{10}$

My Process:

$\cos[\cos^{-1}(\frac{3}{5}) + \frac{\pi}{3}]$

$\cos[\cos^{-1}(\frac{3}{5})] + \cos(\frac{\pi}{3})$

$(\frac{2}{2}) \cdot \frac{3}{5} + \frac{1}{2} \cdot (\frac{5}{5})$

$\frac{6}{10} + \frac{5}{10}$

$\frac{11}{10}$
You cannot separate out the $\cos$ function as you have done in step two.

You can remember this identity.

$$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$

Here $\arccos(\frac{3}{5})$ is an angle ( i.e., approximately $53$ degrees)

Using thus result you should get the desired answer.
Cos^-1(3/5) is 37°. So we get cos(37°+60°) which is equal to -0.39 same as the term you have mentioned as the answer.