Question

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Answer 1

Answer:

8tan^2@-8sec^2@

-8(sec^2@-tan^2@)

-8x1=-8

Answer 2

Question:

Evaluate:

$\displaystyle{\sf\:\dfrac{8}{\cot^2\:\theta}\:-\:\dfrac{8}{\cos^2\:\theta}}$

Answer:

$\displaystyle{\boxed{\red{\sf\:\dfrac{8}{\cot^2\:\theta}\:-\:\dfrac{8}{\cos^2\:\theta}\:=\:-\:8}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:\dfrac{8}{\cot^2\:\theta}\:-\:\dfrac{8}{\cos^2\:\theta}}$

Now,

$\displaystyle{\sf\:\dfrac{8}{\cot^2\:\theta}\:-\:\dfrac{8}{\cos^2\:\theta}}$

$\displaystyle{\implies\sf\:8\:\left(\:\dfrac{1}{\cot^2\:\theta}\:-\:\dfrac{1}{\cos^2\:\theta}\:\right)}$

$\displaystyle{\implies\sf\:8\:\left(\:\tan^2\:\theta\:-\:\dfrac{1}{\cos^2\:\theta}\:\right)\:\:\:-\:-\:-\:\left[\:\because\:\dfrac{1}{\cot^2\:\theta}\:=\:\tan^2\:\theta\:\right]}$

$\displaystyle{\implies\sf\:8\:\left(\:\dfrac{\sin^2\:\theta}{\cos^2\:\theta}\:-\:\dfrac{1}{\cos^2\:\theta}\:\right)\:\:\:-\:-\:-\:\left[\:\because\:\tan^2\:\theta\:=\:\dfrac{\sin^2\:\theta}{\cos^2\:\theta}\:\right]}$

$\displaystyle{\implies\sf\:8\:\left(\:\dfrac{\sin^2\:\theta\:-\:1}{\cos^2\:\theta}\:\right)}$

$\displaystyle{\implies\sf\:8\:\left(\:\dfrac{-\:\cancel{\cos^2\:\theta}}{\cancel{\cos^2\:\theta}}\:\right)\:\:-\:-\:[\:\because\:\sin^2\:\theta\:+\:\cos^2\:\theta\:=\:1\:]}$

$\displaystyle{\implies\sf\:8\:\times\:(\:-\:1\:)}$

$\displaystyle{\implies\sf\:-\:8}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\dfrac{8}{\cot^2\:\theta}\:-\:\dfrac{8}{\cos^2\:\theta}\:=\:-\:8}}}}$