Answer :

So lets get to the problem 

165°= 135° +30° 

To make it easier I'm going to write the same thing like this 
165°= 90° + 45°+30° 

Sin165° 
= Sin ( 90° + 45°+30° ) 
= Cos( 45°+30° )..... (∵ Sin(90 + θ)=cosθ 
= Cos45°Cos30° - Sin45°Sin30° 

Cos165° 
= Cos ( 90° + 45°+30° ) 
= -Sin( 45°+30° )..... (∵Cos(90 + θ)=-Sinθ 
= Sin45°Cos30° + Cos45°Sin30° 


Tan165° 
= Tan ( 90° + 45°+30° ) 
= -Cot( 45°+30° )..... (∵Cot(90 + θ)=-Tanθ 
= -1/tan(45°+30°) 
= -[1-tan45°.Tan30°]/[tan45°+Tan30°] 

Substitute the above values with the following... These should be memorized 
Sin 30° = 1/2 
Cos 30° =[Sqrt(3)]/2 
Tan 30° = 1/[Sqrt(3)] 
Sin45°=Cos45°=1/[Sqrt(2)] 
Tan 45° = 1

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