Answer:

given in explanation

Explanation:

LHS -

[tex]cosec^2A - cot^2A[/tex]

=> [tex]\frac{1}{sin^2A} - \frac{cos^2A}{sin^2A}[/tex]

=> [tex]\frac{1-cos^2A}{sin^2A}[/tex]

=> [tex](sin^2A/sin^2A)[/tex] (since sin^2A + cos^2A = 1)

=> 1 = RHS (Hence, verified)

[tex]sin A = 1/cosec A[/tex]

[tex]cot^2A = cosec^2A - 1[/tex]

=> [tex]cot A =\sqrt{cosec^2A-1}[/tex]

[tex]tan A = \frac{1}{cot A } = \frac{1}{\sqrt{cosec^2A-1} }[/tex]

[tex]cos A = sin A * cos A/sin A = sin A * cot A = 1/cosec A * \sqrt{cosec^2A-1} \\ = \frac{\sqrt{cosec^2A-1} }{cosecA}[/tex]

[tex]sec A = 1/cos A = \frac{cosecA}{\sqrt{cosec^2A-1} }[/tex]