Question

if 5 sin theta =3 find the value of sec²theta-tan²theta/cosec²theta-cot²theta​

Answer 1

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$\: replace \: theta \: by \: \alpha \\ 5sin \alpha = 3 \\ sin \alpha = \frac{3}{5} \\ cosec \alpha = \frac{5}{3} \\ cos \alpha = \sqrt{1 - {sin}^{2} \alpha } \\ cos \alpha = \sqrt{1 - \frac{9}{25} } \\ cos \alpha = \frac{4}{5} \\ sec \alpha = \frac{5}{4} \\ tan \alpha = \frac{sin \alpha }{cos \alpha } = \frac{ \frac{3}{5} }{ \frac{4}{5} } = \frac{3}{4} \\ cot \alpha = \frac{4}{3} \\ now \: \: \frac{ {sec}^{2} \alpha - {tan}^{2} \alpha }{ {cosec}^{2} \alpha - {cot}^{2} \alpha } \\ = \frac{ ({ \frac{5}{4} })^{2} - ({ \frac{3}{4} })^{2} }{( { \frac{5}{3} })^{2} - ({ \frac{4}{3} })^{2} } \\ = \frac{ \frac{25 - 9}{16} }{ \frac{25 - 16}{9} } \\ = \frac{ \frac{16}{16} }{ \frac{9}{9} } \\ = 1$