Question

$if \: sec( \infty ) = \frac{25}{7} then \:find \: the \: value \: of \: \tan( \infty ) and \: \cosec( \infty )$
​

Answer 1

Question :–

$\\ { \bold{if \: \: \sec( \theta ) = \dfrac{25}{7} \:, \: then \: \:find \: \: the \: \: value \: \: \: of \: \tan( \theta ) \: \: and \: cosec( \theta )}}$

ANSWER :–

GIVEN :–

$\\ \: \: \to \: { \bold{ \: \: \sec( \theta ) = \dfrac{25}{7} \: \: }}$

TO FIND :–

$\\ \: \: \to \: { \bold{ \: \: \tan( \theta ) = ? \: \: }}$

$\\ \: \: \to \: { \bold{ \: \: cosec( \theta ) = ? \: \: }}$

SOLUTION :–

$\\ \: \implies { \bold{ \sec( \theta ) = \dfrac{25}{7} \: \: }}$

• We should write this as –

$\\ \: \implies { \bold { \dfrac{1}{\cos( \theta ) }= \dfrac{25}{7} \: \: }}$

$\\ \: \implies { \bold { {\cos( \theta ) }= \dfrac{7}{25} \: \: }}$

• We know that –

$\\ \: \implies { \bold { { \sin}^{2}( \theta) + {\cos {}^{2} ( \theta ) }=1 \: \: }}$

$\\ \: \implies { \bold { { \sin}^{2}( \theta) = 1 - {\cos {}^{2} ( \theta ) }\: \: }}$

$\\ \: \implies { \bold { { \sin}^{}( \theta) = \sqrt{1 - {\cos {}^{2} ( \theta ) }}\: \: }}$

• Now put the value –

$\\ \: \implies { \bold { { \sin}^{}( \theta) = \sqrt{1 - { {( \dfrac{7}{25}) }^{2} }}\: \: }}$

$\\ \: \implies { \bold { { \sin}^{}( \theta) = \sqrt{1 - { { \dfrac{49}{625} }^{} }}\: \: }}$

$\\ \: \implies { \bold { { \sin}^{}( \theta) = \sqrt{{ { \dfrac{625 - 49}{625} }^{} }}\: \: }}$

$\\ \: \implies { \bold { { \sin}^{}( \theta) = \sqrt{{ { \dfrac{576}{625} }^{} }}\: \: }}$

$\\ \: \implies { \bold { { \sin}^{}( \theta) = {{ { \dfrac{24}{25} }^{} }}\: \: }}$

• Now let's find –

$\\ \: \longrightarrow \: { \bold { { \tan}^{}( \theta) = {{ { \dfrac{ \sin( \theta) }{ \cos( \theta) } }^{} }}\: \: }}$

$\\ \: \implies \: { \bold { { \tan}^{}( \theta) = {{ { \dfrac{ \dfrac{24}{ \cancel{25}} }{ \dfrac{7}{ \cancel{25}} } }^{} }}\: \: }}$

$\\ \: \implies \: \large { \boxed{ \bold { { \tan}^{}( \theta) = {{ { \dfrac{ 24 }{ 7 } }^{} }}\: \: }}}$

$\\ \: \longrightarrow \: { \bold { { cosec}^{}( \theta) = {{ { \dfrac{ 1}{ \sin( \theta) } }^{} }}\: \: }}$

$\\ \: \implies \: { \bold { { cosec}^{}( \theta) = {{ { \dfrac{ 1}{ \dfrac{24}{25} } }^{} }}\: \: }}$

$\\ \: \implies \: \large{ \boxed { \bold { { cosec}^{}( \theta) = {{ { \dfrac{ 25}{ 24 } }^{} }}}\: \: }}$