Question #ce4ab

1 Answer
Oct 15, 2017

One cannot prove this as an identity, because it is only true for these values of #theta#:

#theta = pi/2 + npi# and #theta = pi/4 + npi#; #n in ZZ#

Explanation:

If you want the values for #theta#, then starting with:

#cot(theta)cos(theta)csc(theta)= cot(theta)#

Use the identity #csc(theta) = 1/sin(theta)#:

#cot(theta)cos(theta)/sin(theta)= cot(theta)#

Use the identity #cos(theta)/sin(theta)= cot(theta)#

#cot^2(theta) = cot(theta)#

Subtract #cot(theta)# from both sides:

#cot^2(theta) - cot(theta)= 0#

Factor:

#cot(theta)(cot(theta) - 1)= 0#

There are two roots:

#cot(theta) = 0 and cot(theta) = 1#

Use the inverse cotangent on both:

#theta = cot^-1(0) and theta = cot^-1(1)#

The primary values for these are:

#theta = pi/2# and #theta = pi/4#

The cotangent function will repeat the roots at integer multiples of #pi#:

#theta = pi/2 + npi# and #theta = pi/4 + npi#; #n in ZZ#