$\"\"$

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Mean value theorem :

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Let $\"\"$ be a function that satisfies the following three hypotheses.

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1. $\"\"$ is continuous on $\"\"$ .

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2. $\"\"$ is differentiable on $\"\"$ .

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Then there is a number $\"\"$ in $\"\"$ such that $\"\"$ .

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$\"\"$

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Consider the function is $\"\"$ .

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The function $\"\"$ is continuous on $\"\"$ .

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Differentiate $\"\"$ on each side with respect to $\"\"$ .

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$\"\"$ .

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The function $\"\"$ is differentiable on $\"\"$ .

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Then $\"\"$ .

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The range of the cosine function is $\"\"$ .

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$\"\"$

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From the mean value theorem,

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$\"\"$

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Substitute $\"\"$ in above expression.

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$\"\"$

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$\"\"$

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Substitute $\"\"$ in above expression.

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$\"\"$

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$\"\"$

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$\"\"$ .