Verify the identity. cotangent of x to the second power divided by quantity cosecant of x plus one equals quantity one minus sine of x divided by sine of x

Question

Precalculus

Answer 1

(cot x)^2/(csc x + 1) =  (1 - sin x)/sin x

 

Left hand side:

   

  denominator is:

 

       1/sinx + 1 = (1 + sinx)/sin x

 

       The reciprocal is sin x / (1 + sinx)

 

 

The numerator is  cot x^2 = (cosx)^2 / (sinx)^2

 

                  = ( 1 - (sinx)^2)/(sinx)^2

 

                  = ( 1 + sinx)(1-sinx)/(sinx)^2

 

Multiplying them together:

 

  ( 1+ sinx)(1-sinx)/(sinx)^2 * sinx/(1+sinx)

 

 (1-sinx)/sinx <--- factor of 1+sinx cancels, as does 1 factor of sine

 

This expression is identical to the right hand side, which concludes the proof