Question

if cosecθ-cotθ=q then the value of 2cosec^2θ+3cot^2θ

Answer 1

Answer:

$\huge{ \pink{\frac{ {q}^{4} - 2 {q}^{2} + 5}{4 {q}^{2} } }}$

Step-by-step explanation:

Given that

${1 \over \sin(x) }{- \frac{ \cos(x) }{ \sin(x) } } = q \\ \\ \frac{2 \sin( \frac{x}{2} ) }{2 \sin {}^{2} ( \frac{x}{2} )\cos( \frac{x}{2} ) } = q \\ \\ \tan( \frac{x}{2 } ) = q$

Now

$2 \csc {}^{2} (x) + 3 \cot {}^{2} (x) \\ \\ \frac{2}{ \sin {}^{2} (x) } + \frac{3 \cos { }^{2} (x) }{ \sin {}^{2} (x) } \\ \\ = \frac{2 + 3(1 - \sin {}^{2} (x)) }{ \sin {}^{2} (x) } \\ \\ = \frac{5 - 3\sin {}^{2} (x) }{ \sin {}^{2} (x) }$

Now

$\sin(x) = \frac{2 \tan( \frac{x}{2} ) }{1 + \tan {}^{2} ( \frac{x}{2} ) } \\ \\ \\ \sin(x) = \frac{2q}{1 + {q}^{2} }$

Put value of sin x

$\frac{5 - 3(\frac{2q}{1 + {q}^{2} }) {}^{2} }{(\frac{2q}{1 + {q}^{2} }) {}^{2} } \\ \\ \\ = \frac{5(1 + {q}^{2} ) {}^{2} - 12 {q}^{2} }{4 {q}^{2} } \\ \\ = \frac{ {q}^{4} + 10 {q}^{2} + 5 - 12 {q}^{2} }{4 {q}^{2} } \\ \\ = \frac{ {q}^{4} - 2 {q}^{2} + 5}{4 {q}^{2} }$