Question

if cosec a +cot a =q
prove that cosec a - cot a =1/q
and hence find values of sin a and sec a

Answer 1

Given : cosec a+cot a = q ---------> 1

We know that cosec²Ф-cot²Ф = 1

                       (cosec a+cot a)+(cosec a-cot a) = 1

                       cosec a-cot a = 1/(cosec a+cot a)

                       cosec a-cot a = 1/q  -------> 2

                 ∴ Hence proved.

Equation 1 & Equation 2

                      cosec a+cot a = q

                      cosec a-cot a = 1/q

                  -----------------------------------

                      2cosec a +0 = q+1/q

                     2cosec a = (q²+1)/q

                     cosec a = (q²+1)/2q

                     (1/sin a) = (q²+1)/2q

                     sin a = 2q/(q²+1)

          ∴sin a = 2q/(q²+1)

sec a = ?

We know that sin²Ф+cos²Ф = 1

                       [2q/(q²+1)]²+cos²Ф =1

                       cos² a = 1-[2q/(q²+1)]²

                       cos² a = 1-4q²/(q²+1)²

                       cos² a = [(q²)²+1²+2.q².1-4q²]/(q²+1)²

                      cos² a = [(q²)²+1²-2.q².1]/(q²+1)²

                      cos² a = (q²-1)/(q²+1)

          ∴cos² a = (q²-1)/(q²+1)