Simplify #("csc"^2x)/(1 + tan^2x)# ?

1 Answer

Answer:

See below

Explanation:

The easiest trigonometric ratio that it could simplify to is #cot^2x#

Step1: Convert everything to #sinx# and #cosx#

#csc^2x# can be converted into #1/sin^2x# according to the reciprocal identities and #1+tan^2x# can be converted into #sec^2x# according to the Pythagorean identities, and then convert that into #1/cos^2x# because its the reciprocal.

Step2: Plug these converted identities back into the equation

#(1/sin^2x)/(1/cos^2x)#

when dividing by a fraction, you flip the bottom fraction and multiply.

Answer

#(1/sin^2x) ⋅ (cos^2x/1) = cos^2x/sin^2x = cot^2x#

according to the Quotient identities.