#tantheta - cot theta = 0#
Rewrite using the identities #tantheta = sintheta/costheta# and #cottheta = costheta/sintheta#.
#sintheta/costheta - costheta/sintheta = 0#
Simplify the fraction
#(sin^2theta - cos^2theta)/(sinthetacostheta)= 0#
Multiply both sides of the equation by #sinthetacostheta# to remove the denominator
#sin^2theta - cos^2theta = 0#
Replace #sin^2theta# with #1-cos^2theta# by rearranging this identity: #sin^2theta+cos^2theta-=1#
#1 - cos^2theta - cos^2theta = 0#
#1 - 2cos^2theta = 0#
#-2cos^2theta= -1#
#cos^2theta = 1/2#
#costheta = +-1/sqrt(2)#
#theta=cos^-1(+-1/sqrt2)=45, 135#
These values are the principle values of #theta#, but we need to find all the values of #theta# that satisfy this equation within our given range.
To find the complementary angles, where #y# is #+"ve"#, within one period of the cosinusoidal wave, we need to subtract #theta# from #360#: #a=360-45=315#
To find the complementary angles, where #y# is #-"ve"#, within one period of the cosinusoidal wave, we need to add #theta# to #90#: #b=90+135=225#
So all the values of #theta# within the given range are:
#theta= 45, 135, 225 " and " 315#
Another way to find the other values of theta is this relation:
#theta_n=theta+n(270), costheta>=0#
#theta_n=theta+n(90), costheta<0#
Note that #n# must be an integer. Note that we use #n(360)# only when dealing with the sine and cosine functions; when dealing with the tangent, we use the period #n(180)#.
