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Simplify the expression : $ab*\arctan\left(\frac{\cot x}{b}\right)$

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Evaluate rational exponents

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Q. Simplify the expression :

ab*\arctan\left(\frac{\cot x}{b}\right)

Rewrite using identities: Step $1$ : Rewrite the expression using trigonometric identities. $\newline$ Arctan and cotangent are inverse functions, so $\text{acot}(x)$ can be rewritten as $\frac{1}{\tan(x)}$ . Thus, the expression becomes $ab\text{Arctan}\left(\frac{\frac{1}{\tan(x)}}{b}\right)$ .
Simplify inside Arctan: Step $2$ : Simplify the expression inside Arctan. $\newline$ $(\frac{1}{\tan(x)})/b$ simplifies to $\frac{1}{b\tan(x)}$ . Therefore, the expression now is $ab\text{Arctan}(\frac{1}{b\tan(x)})$ .
Apply Arctan property: Step $3$ : Apply the property of Arctan. $\text{Arctan}(\frac{1}{y}) = \cot(y)$ for $y = b\tan(x)$ . Therefore, $ab\text{Arctan}(\frac{1}{(b\tan(x))})$ simplifies to $ab\cot(b\tan(x))$ .

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