Question

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

Answer 1

For principal values of the angles, the answer is `33/45`. For general values there are two values, `+-33/45`.

Explanation:

Let a = arc tan (5/12). `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 =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`

`=(12/13)(4/5)-(5/13)(3/5),`(for principal values of angles)

`=33/65`.

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

are `+-33/65`