How do I prove and find the domain of the following trig identity?

(cos theta-sin theta)/(cos theta +sin theta)= (cot theta-1)/(cot theta + 1)

1 Answer
Nov 22, 2016

see below

Explanation:

(cos theta - sin theta)/(cos theta+sin theta)=(cot theta-1)/(cot theta+1)

Right Side : =(cot theta-1)/(cot theta+1)

=(cos theta/sin theta-1)/(cos theta/sin theta+1)

=((cos theta-sin theta)/sin theta)/((cos theta+sin theta)/sin theta

=(cos theta-sin theta)/sin theta * sin theta/(cos theta+sin theta)

=(cos theta-sin theta)/cancel sin theta * cancel sin theta/(cos theta+sin theta)

=(cos theta-sin theta)/(cos theta+sin theta)

:.= Left Side

Domain :
The denominator cannot be zero. So take the denominator set it equal to zero and then solve.

cos theta+sin theta=0

Use linear combination to solve the equation

A=1, B=1, C= sqrt 2. Note that this is in quadrant one since both cosine and sine are positive

cos D=A/C = 1/sqrt2

D=cos^-1(1/sqrt2)=45^@

sqrt2cos(theta-45^@)=0--> Put it in the form C cos (theta-D)=0

cos(theta-45^@)=0

theta-45^@=cos^-1 0

theta-45^@=+-90^@ +360^@ n

theta=45^@+-90^@ +360^@ n

theta=135^@ +360^@ n or theta=-45^@ + 360^@ n

where n=0,+-1,+-2,+-3,...

Therefore,

D:{theta inR,theta !=135^@ +360^@ n,theta!=-45^@ + 360^@ n, n=0,+-1,+-2,+-3,...}