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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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