How do you find the exact value of $cos [a r c tan (\frac{5}{12}) + a r c cot (\frac{4}{3})$ $cos [arc tan ( 5/12 ) + arc cot ( 4/3 )$ ?

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Answer 1 · 2016-07-04T16:19:16

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

Let a = arc tan (5/12). $tan a = \frac{5}{12} > 0$ $tan a =5/12>0$ . 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). $cot b = \frac{4}{3} > 0$ $cot b =4/3>0$ . 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
$cos (a + b) = cos a cos b - sin a sin b$ $cos ( a + b )=cos a cos b - sin a sin b$

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

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

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

are $\pm \frac{33}{65}$ $+-33/65$

How do you find the exact value of cos[arctan(512)+arccot(43)cos [arc tan ( 5/12 ) + arc cot ( 4/3 )?