# How do you find the exact value of cos [arc tan ( 5/12 ) + arc cot ( 4/3 )?

Jul 4, 2016

For principal values of the angles, the answer is $\frac{33}{45}$. For general values there are two values, $\pm \frac{33}{45}$.

#### Explanation:

Let a = arc tan (5/12). $\tan a = \frac{5}{12} > 0$. The principal a is in the 1st

quadrant. So, cos a=12/13 and sin a = 5/13. The general values are

in 1st and 3rd. For general values, both cos and sin have the same

sign..

Let b = arc cot (4/3). $\cot b = \frac{4}{3} > 0$. The principal b is in the 1st

quadrant. So, cos b=4/5 and sin b =3/5. The general values are in

1st and 3rd. For general values, both cos and sin have the same

sign.

Now, the given expression is
$\cos \left(a + b\right) = \cos a \cos b - \sin a \sin b$

$= \left(\frac{12}{13}\right) \left(\frac{4}{5}\right) - \left(\frac{5}{13}\right) \left(\frac{3}{5}\right) ,$(for principal values of angles)

$= \frac{33}{65}$.

Considering same sign for both sin and cos, the general values

are $\pm \frac{33}{65}$