Question

Hlw Friend's ☺️❤️

Anyone solve this question ☺️

Answer 1

$\underline{\mathfrak{Solution : }}$

$\textsf{Trigonometric identities to be used,} \\ \\ \\ \mathsf{\implies {tan}^{2}\theta \: = \: \dfrac{{sin}^{2}\theta}{{cos}^{2}\theta}}\\ \\ \\ \mathsf{\implies {cosec}^{2}\theta \: = \: \dfrac{1}{{sin}^{2}\theta}} \\ \\ \\ \mathsf{\implies {sec}^{2}\theta \: = \: \dfrac{1}{{cos}^{2}\theta}}$


$\textsf{Given,} \\ \\ \mathsf{\implies RHS \: = \: \dfrac{1}{{sin}^{2} \theta \: - \: {cos}^{2}\theta}} \\ \\ \\ \\ \\ \mathsf{\implies LHS \: = \: \dfrac{{tan}^{2}\theta}{{tan}^{2}\theta \: - \: 1} \: + \: \dfrac{{cosec}^{2}\theta}{{sec}^{2}\theta \: - \: {cosec}^{2}\theta}}$


$\mathsf{= \: \dfrac{\quad \dfrac{sin^{2}\theta}{cos^{2}\theta} \quad}{ \quad\dfrac{sin^{2}\theta}{cos^{2}\theta} \: - \: 1 } \quad \: + \: \dfrac{ \quad \dfrac{1}{ {sin}^{2} \theta } \quad}{ \quad \dfrac{1}{ {cos}^{2} \theta} \: - \: \dfrac{1}{ {sin}^{2} \theta} }}$


$\mathsf{ = \: \dfrac{ \quad\dfrac{ {sin}^{2} \theta}{ \cancel{{cos}^{2} } \theta} \quad}{ \quad \dfrac{ {sin}^{2} \theta \: - \: {cos}^{2} \theta }{ \cancel{ {cos}^{2} \theta} } \quad} \: + \: \dfrac{ \quad \dfrac{1}{ \cancel{ {sin}^{2} \theta }} \quad}{ \quad \dfrac{ {sin}^{2} \theta \: - \: {cos}^{2} \theta }{ \cancel{{sin}^{2} \theta }\: {cos}^{2} \theta}} }$


$\\ \mathsf{ = \: \dfrac{ {sin}^{2} \theta}{ \quad{sin}^{2} \theta \: - \: {cos}^{2} \theta \quad} \: + \: (1 \: \div \: \dfrac{ {sin}^{2} \theta \: - \: {cos}^{2} \theta}{ {cos}^{2} \theta } )}$


$\\ \mathsf{ = \: \dfrac{ {sin}^{2} \theta}{ \quad {sin}^{2} \theta \: - \: {cos}^{2} \theta \quad } \: + \: \dfrac{ {cos}^{2} \theta }{ \quad {sin}^{2} \theta \: - \: {cos}^{2} \theta \quad }}$


$\mathsf{ = \: \dfrac{ {sin}^{2} \theta \: + \: {cos}^{2} \theta }{ \quad {sin}^{2} \theta \: - \: {cos}^{2} \theta \quad }} \\ \\ \\\mathsf{Using \: trigonometric \: identity : } \\ \\ \\ \boxed{ \mathsf{ \implies {sin}^{2} \theta \: + \: {cos}^{2} \theta \: = \: 1 }} \\ \\ \\ \mathsf{ \implies \dfrac{1}{ \quad {sin}^{2} \theta \: - \: {cos}^{2} \theta \quad } \: = \: \boxed{ \mathsf{RHS}}}$



$\boxed{\underline{ \mathsf{Proved !! }}}$

Answer 2

Hope it may help u ...

Name Harshit