Question

( 1 - tan thetha ) ^2 + ( 1 - cot tha ) ^2 = ( sec thetha - cosec thetha ) ^2

Answer 1

your answer is in giving picture

Answer 2

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