What does #4sin(arc tan(-7))-2cot(arccot(4))# equal?

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Answer 1 · 2016-07-27T05:06:17

#=-113/(17 sqrt 50)#

Let #a = arc tan (-7) in (-pi/2, 0)#.

Then, #sin a =-7/sqrt 50 and cos a=1/sqrt 50#.

Let #b = arc cot(4) in (0, pi/2)#.

Also, #2b in (0, pi/2)#

Then, #sin b= 1/sqrt 17, cos b = 4/sqrt 17#,

#sin 2b = 2 sin b cos b = 8/17# and

#cos 2b =2 cos^2 b - 1 = 15/17#.

Now, the given expression is

#4 sin (a - 2b)#

#=4(sina cos 2b- cos a sin 2b)#

#=4((-7/sqrt 50)(15/17)-(1/sqrt 50)(8/17)#

#=-113/(17 sqrt 50)#.