Expert: Ahmed Salami - 10/29/2008

Question
1/tan squared x- 1/cot squared x=csc squared x - sex squared x

Answer
Hi Lorin,
Lets start with the identity
cos^2(x) + sin^2(x) = 1
dividing through by cos^2(x), we have
cos^2(x)/cos^2(x) + sin^2(x)/cos^2(x) = 1/cos^2(x)
1 + (sinx/cosx)^2 = (1/cosx)^2
1 + tan^2(x) = sec^2(x)
i.e
tan^2(x) = sec^2(x) - 1
going back again to cos^2(x) + sin^2(x) = 1
and dividing through this time by sin^2(x), we have
cos^2(x)/sin^2(x) + sin^2(x)/sin^2(x) = 1/sin^2(x)
(cosx/sinx)^2 + 1 = (1/sinx)^2
cot^2(x) + 1 = cosec^2(x)
i.e
cot^2(x) = cosec^2(x) - 1
But
1/tan^2(x) - 1/cot^2(x)
= cot^2(x) - tan^2(x)
= {cosec^2(x) - 1} - {sec^2(x) - 1}
= cosec^2(x) - 1 - sec^2(x) + 1
= cosec^2(x) - sec^2(x)

Regards.