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## Sum and Difference Trigonometric Formulas

These formulas explain how to add and subtract trigonometric functions (and their arguments). If you've got sum time, see what a difference these formulas will make for your trig toolkit.

# Inverse Trigonometric Identities

Given that $$\cos ( \arcsin \frac {12}{13} - \arctan \frac {16}{12} ) = \frac {a }{b}$$ where $$a$$ and $$b$$ are positive coprime integers, what is the value of $$a+b$$?

Let $$x = \sin\left(\sin^{-1} \left(\frac{3}{5}\right) + \tan^{-1} (2)\right)$$. $$x$$ can be written in the form $$\frac{a\sqrt{b}}{c}$$, where $$a, b$$ and $$c$$ are positive integers, $$a$$ and $$c$$ are coprime and $$b$$ is not divisible by the square of any prime. What is the value of $$a+b+c$$?

Details and assumptions

$$\sin^{-1}$$ and $$\tan^{-1}$$ represent the inverse of the $$\sin$$ and $$\tan$$ function, and not the reciprocal.

Which of the following is equal to

$\arccos ( - x) ?$

If $$x \in (0, \pi )$$, which of the following is equal to

$\arctan ( \cot x) ?$

$\cos^{-1} \frac{a}{b} = 2 \tan^{-1} \frac{ \sqrt{72}} {12 },$ where $$a$$ and $$b$$ are positive, coprime integers. What is $$a+b$$?

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