Question

tan2 theta/tan2theta-1 + cosec2 theta /sec2 theta - cosec2 theta = 1/sin2 theta - cos2 theta
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Answer 1

$\huge\underline\bold\red{★Answer★}$

$\frac{ {tan}^{2} \theta }{ {tan}^{2} \theta - 1 } + \frac{ {cosec}^{2} \theta }{ {sec}^{2} \theta - {cosec}^{2} \theta } \\ = \frac{ \frac{ {sin}^{2} \theta }{ {cos}^{2} \theta } }{ \frac{ {sin}^{2} \theta - {cos}^{2} \theta }{ {cos}^{2} \theta } } + \frac{ \frac{1}{ {sin}^{2} \theta } }{ \frac{1}{ {cos}^{2} \theta } - \frac{1}{ {sin}^{2} \theta } } \\ = \frac{ {sin}^{2} \theta }{ {sin}^{2} \theta - {cos}^{2} \theta } + \frac{1}{ {sin}^{2} \theta } \times \frac{ {sin}^{2} \theta {cos}^{2} \theta }{ {sin}^{2} \theta - {cos}^{2} \theta } \\ = \frac{ {sin}^{2} \theta }{ {sin}^{2} \theta - {cos}^{2} \theta } + \frac{ {cos}^{2} \theta }{ {sin}^{2} \theta - {cos}^{2} \theta } \\ = \frac{ {sin}^{2} \theta + {cos}^{2} \theta }{ {sin}^{2} \theta - {cos}^{2} \theta } \\ = \frac{1}{ {sin}^{2} \theta - {cos}^{2} \theta }$