Question

(tan25/cosec65)²+( cot25/sec65)²=?​

Answer 1

$({ \frac{ tan25 }{ cosec \: 65 } })^{2} + ( { \frac{cot \: 25}{sec \: 65} })^{2} \\ =( { \frac{tan(90 - 65)}{cosec \: 65} })^{2} + ( { \frac{cot(90 - 65)}{sec \: 65} })^{2} \\ =( { \frac{cot \: 65}{cosec \: 65} })^{2} + ( { \frac{tan \: 65}{sec \: 65} })^{2} \\ = ( { \frac{cos \: 65}{sin \: 65} \times sin \: 65})^{2} + ( { \frac{sin \: 65}{cos \: 65} \times cos \: 65})^{2} \\ = {cos}^{2} 65 + {sin}^{2} 65 \\ = 1$

Hence, your answer is 1.

Some of the identities are used here which I had written in the attachment.

So, go through the attachment also as it would be helpful.