Question

sec^2theta . cos^2 theta =​

Answer 1

Answer:

sec^2 theta . cos^2 theta = 1

Answer 2

Answer:

$\displaystyle{\boxed{\red{\sf\:\sec^2\:\theta\:.\:\cos^2\:\theta\:=\:1\:}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:\sec^2\:\theta\:.\:\cos^2\:\theta\:}$

We have to find the value of this expression.

Now,

$\displaystyle{\sf\:\sec^2\:\theta\:.\:\cos^2\:\theta\:}$

$\displaystyle{\implies\sf\:\dfrac{1}{\cos^2\:\theta}\:\times\:\cos^2\:\theta\:\quad\:-\:-\:-\:\left[\:\because\:\sec\:\theta\:=\:\dfrac{1}{\cos\:\theta}\:\right]}$

$\displaystyle{\implies\sf\:\dfrac{1}{\cancel{\cos^2\:\theta}}\:\times\:\cancel{\cos^2\:\theta}}$

$\displaystyle{\implies\sf\:1\:\times\:1}$

$\displaystyle{\implies\sf\:1}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\sec^2\:\theta\:.\:\cos^2\:\theta\:=\:1\:}}}}$

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Alternative Method:

We have given that,

$\displaystyle{\sf\:\sec^2\:\theta\:.\:\cos^2\:\theta\:}$

Now,

$\displaystyle{\sf\:\sec^2\:\theta\:.\:\cos^2\:\theta\:}$

$\displaystyle{\implies\sf\:(\:1\:+\:\tan^2\:\theta\:)\:\times\:\cos^2\:\theta\:\quad\:-\:-\:-\:[\:\sec^2\:\theta\:=\:1\:+\:\tan^2\:\theta\:]}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:+\:\cos^2\:\theta\:\times\:\tan^2\:\theta}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:+\:\cancel{\cos^2\:\theta}\:\times\:\dfrac{\sin^2\:\theta}{\cancel{\cos^2\:\theta}}\:\quad\:-\:-\:-\:\left[\:\because\:\tan\:\theta\:=\:\dfrac{\sin\:\theta}{\cos\:\theta}\:\right]}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:+\:\sin^2\:\theta}$

$\displaystyle{\implies\sf\:1}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\sec^2\:\theta\:.\:\cos^2\:\theta\:=\:1\:}}}}$