Question

Calculus: tan (2 arc cos (x/3)) = ?

Answer 1

What are you supposed to do? Integrate, derive, solve for x...?

Answer 2

it says to simplify the expression

Answer 3

Since we know arccos(x/3) gives us an angle \(\alpha\) whose cosine is x/3, we can draw the following triangle and use Pytharogus to calculate the length of the third side:

From this we get:\[\tan(\alpha)=\frac{\sqrt{9-x^2}}{x}\tag{a}\]Now just use the formula for \(\tan(2\alpha)\) to get:\[\tan(2\alpha)=\frac{2\tan(\alpha)}{1-\tan^2(\alpha)}\tag{b}\]Substitute equation (a) into equation (b) and simplify.