Question #bca2e

2 Answers
Mar 28, 2017

It doesn't. A counterexample is #x=pi/6#

Explanation:

One form of the Pythagorean Identity states that #cot^2x+1=csc^2x#

If #cot^2x+sin^2x=csc^2x# then by the transitive property, #cot^2x+sin^2x=cot^2x+1#

Subtracting #cot^2x# from both sides,
#sin^2x=1#

If, say, #x=pi/6# then #sin^2x=1/4!=1#, and since we can provide a counterexample the identity is not true.

Plugging this counterexample into the original identity, we get #cot^2(pi/6)+sin^2(pi/6)=3.25# on the left side and #csc^2(pi/6)=4#. Obviously #3.25!=4# so the identity cannot be true.

Mar 28, 2017

#sin^2x=1# in the end, hence both sides of the equation are accounted for.
Read on for more info...

Explanation:

#cot^2x+sin^2x=csc^2x#

Since #csc^2x=1+cot^2x#,
#cot^2x+sin^2x=1+cot^2x#
#cot^2x-cot^2x+sin^2x=1#
#sin^2x=1#

Therefore,
#cot^2x+sin^2x=cot^2x+1=csc^2x#