Sum and Difference Trigonometric Formulas

Inverse Trigonometric Identities


Given that cos(arcsin1213arctan1612)=ab \cos ( \arcsin \frac {12}{13} - \arctan \frac {16}{12} ) = \frac {a }{b} where aa and bb are positive coprime integers, what is the value of a+ba+b ?

Let x=sin(sin1(35)+tan1(2))x = \sin\left(\sin^{-1} \left(\frac{3}{5}\right) + \tan^{-1} (2)\right). xx can be written in the form abc\frac{a\sqrt{b}}{c}, where a,ba, b and cc are positive integers, aa and cc are coprime and bb is not divisible by the square of any prime. What is the value of a+b+ca+b+c?

Details and assumptions

sin1\sin^{-1} and tan1\tan^{-1} represent the inverse of the sin\sin and tan\tan function, and not the reciprocal.

Which of the following is equal to

arccos(x)? \arccos ( - x) ?

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

arctan(cotx)? \arctan ( \cot x) ?

cos1ab=2tan17212, \cos^{-1} \frac{a}{b} = 2 \tan^{-1} \frac{ \sqrt{72}} {12 }, where aa and bb are positive, coprime integers. What is a+ba+b?


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