Question

pls solve this question ASAP​

Answer 1

Question:

If $\displaystyle{\sf\:\sec\:\theta\:=\:\dfrac{3}{2}}$ , then the value of $\displaystyle{\sf\:2\:\cos^2\:\theta\:+\:2\:\cot^2\:\theta\:-\:9}$ is

$\displaystyle{\sf\:a)\:\dfrac{-\:91}{15}}$

$\displaystyle{\sf\:b)\:\dfrac{91}{15}}$

$\displaystyle{\sf\:c)\:\dfrac{44}{15}}$

$\displaystyle{\sf\:d)\:\dfrac{90}{44}}$

Answer:

$\displaystyle{\boxed{\red{\sf\:2\:\cos^2\:\theta\:+\:2\:\cot^2\:\theta\:-\:9\:=\:\dfrac{-\:293}{45}\:}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:\sec\:\theta\:=\:\dfrac{3}{2}}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:\cos\:\theta\:=\:\dfrac{1}{\sec\:\theta}\:}}}$

$\displaystyle{\implies\sf\:\cos\:\theta\:=\:\dfrac{1}{\dfrac{3}{2}}}$

$\displaystyle{\implies\sf\:\cos\:\theta\:=\:\dfrac{2}{3}}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:=\:\left(\:\dfrac{2}{3}\:\right)^2}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:=\:\dfrac{2^2}{3^2}}$

$\displaystyle{\implies\:\boxed{\blue{\sf\:\cos^2\:\theta\:=\:\dfrac{4}{9}\:}}}$

Now, we know that,

$\displaystyle{\boxed{\blue{\sf\:\sec^2\:\theta\:=\:1\:+\:\tan^2\:\theta\:}}}$

$\displaystyle{\implies\sf\:\tan^2\:\theta\:=\:\sec^2\:\theta\:-\:1}$

$\displaystyle{\implies\sf\:\tan^2\:\theta\:=\:\left(\:\dfrac{3}{2}\:\right)^2\:-\:1}$

$\displaystyle{\implies\sf\:\tan^2\:\theta\:=\:\dfrac{3^2}{2^2}\:-\:1}$

$\displaystyle{\implies\sf\:\tan^2\:\theta\:=\:\dfrac{9}{4}\:-\:1}$

$\displaystyle{\implies\sf\:\tan^2\:\theta\:=\:\dfrac{9\:-\:4}{4}}$

$\displaystyle{\implies\:\boxed{\orange{\sf\:\tan^2\:\theta\:=\:\dfrac{5}{4}\:}}}$

We know that,

$\displaystyle{\boxed{\green{\sf\:\cot\:\theta\:=\:\dfrac{1}{\tan\:\theta}\:}}}$

$\displaystyle{\implies\sf\:\cot^2\:\theta\:=\:\dfrac{1}{\tan^2\:\theta}}$

$\displaystyle{\implies\sf\:\cot^2\:\theta\:=\:\dfrac{1}{\dfrac{5}{4}}}$

$\displaystyle{\implies\:\boxed{\purple{\sf\:\cot^2\:\theta\:=\:\dfrac{4}{5}\:}}}$

Now, we have to find the value of

$\displaystyle{\sf\:2\:\cos^2\:\theta\:+\:2\:\cot^2\:\theta\:-\:9}$

$\displaystyle{\implies\sf\:2\:\times\:\dfrac{4}{9}\:+\:2\:\times\:\dfrac{4}{5}\:-\:9}$

$\displaystyle{\implies\sf\:\dfrac{8}{9}\:+\:\dfrac{8}{5}\:-\:9}$

$\displaystyle{\implies\sf\:\dfrac{8\:\times\:5\:+\:8\:\times\:9}{9\:\times\:5}\:-\:9}$

$\displaystyle{\implies\sf\:\dfrac{40\:+\:72}{45}\:-\:9}$

$\displaystyle{\implies\sf\:\dfrac{112}{45}\:-\:9}$

$\displaystyle{\implies\sf\:\dfrac{112\:-\:9\:\times\:45}{45}}$

$\displaystyle{\implies\sf\:\dfrac{112\:-\:405}{45}}$

$\displaystyle{\implies\sf\:\dfrac{-\:293}{45}}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:2\:\cos^2\:\theta\:+\:2\:\cot^2\:\theta\:-\:9\:=\:\dfrac{-\:293}{45}\:}}}}$