Question

prove that (1+tan^2theta)+(1+1/tan^2theta)=1/sin^2theta-sin^4theta​

Answer 1

$(1 + {tan}^{2} \theta) + (1 + \frac{1}{ {tan}^{2} \theta} ) \\ = {sec}^{2} \theta + (1 + {cot}^{2} \theta) \\ = {sec}^{2} \theta + {cosec}^{2} \theta \\ = \frac{1}{ {cos}^{2} \theta} + \frac{1}{ {sin}^{2} \theta} \\ = \frac{1}{ {sin}^{2} \theta {cos}^{2} \theta } \\ = \frac{1}{ {sin}^{2} \theta(1 - {sin}^{2} \theta) } \\ = \frac{1}{ {sin}^{2} \theta - {sin}^{4} \theta }$