cos(x)/sin^2(x) - cos(x)*cot^2(x) = cot(x)*sin(x)
since cot(x) = cos(x)/sin(x), right side of this equation becomes:
cos(x)/sin(x)*sin(x) = cos(x)
your equation becomes:
cos(x)/sin^2(x) - cos(x)*cot^2(x) = cos(x)
since cot^2(x) is equal to cos^2(x)/sin^2(x), the left side of your equation becomes:
cos(x)/sin^2(x) - cos(x)*cos^2(x)/sin^2(x)
since the denominator are now the same, this expression becomes:
[cos(x) - cos(x)*cos^2(x)]/sin^2(x)
factor out the cos(x) in the numerator to get:
[cos(x)*(1-cos^2(x)]/sin^2(x)
since 1-cos^2(x) is equal to sin^2(x), this expression becomes:
[cos(x)*sin^2(x)]/sin^2(x)
the sin^2(x) in the numerator and the sin^2(x) in the denominator cancel out and you are left with:
cos(x) on the left side of your equation.
since cos(x) is on the right side of your equation, the equation becomes:
cos(x) = cos(x) proving the identity.
the following piece of paper shos the same calculations done manually which should be easier to follow:
number 1 shows the original equation to be solved.
number 2 shows the basic identities that are used in the proof.
number 3 replaces cot^2(x) with cos^2(x)/sin^2(x) and replaces cot(x) with cos(x)/sin(x)
number 4 simplifies the equation in number 3.
number 5 factors out cos(x) that is common to the 2 terms in the numerator or the expression on the left side of the equation.
number 6 simplifies the equation from number 5.
number 7 shows the final result which proves the identity is true.
