(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,...}