Sum and Difference Trigonometric Formulas
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 \(a+b\)?