How do you find the exact value of #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 #33/45#. For general values there are two values, #+-33/45#.

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#