How can I solve this without using a calculator?

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Answer 1 · 2018-01-02T22:50:02

See below.

#cot(x)=1/tan(x)=cos(x)/sin(x)#

This uses the following identities:

#color(red)(cos(A-B)=cosAcosB+sinAsinB)#

#color(red)(sin(A-B)=sinAcosA-cosAsinB)#

#cos(-(5pi)/4)=cos((-3pi)/4-(2pi)/4)=#

#->cos(-(3pi)/4)cos(-(2pi)/4)+sin(-(3pi)/4)sin(-(2pi)/4)#

#->-sqrt(2)/2(0)+(-sqrt(2)/2(-1))=sqrt(2)/2#

#sin(-(5pi)/4)=sin(-(3pi)/4-(2pi)/4)=#

#sin(-(3pi)/4)cos(-(2pi)/4)-cos(-(3pi)/4)sin(-(2pi)/4)#

#(-sqrt(2)/2)(0)-((-sqrt(2)/2)(-1))=-sqrt(2)/2#

#(cos(-(5pi)/4))/sin(-(5pi)/4)=(sqrt(2)/2)/(-sqrt(2)/2)=-1#