First slide

Compound angles

Question

Let $f (θ) = \frac{\cot θ}{1 + \cot θ} and α + β = \frac{5 π}{4}$ then the value of $f (α) f (β)$ is

Moderate

A

$\frac{1}{2}$
B

$- \frac{1}{2}$
C

2
D

none of these

Solution

$\begin{matrix} f (β) = f (\frac{5 π}{4} - α) = \frac{\cot (\frac{5 π}{4} - α)}{1 + \cot (\frac{5 π}{4} - α)} \\ = \frac{1}{1 + \tan (\frac{5 π}{4} - α)} \\ = \frac{1}{1 + \frac{1 - \tan α}{1 + \tan α}} = \frac{1 + \tan α}{2} \end{matrix}$
$\begin{array}{l} as, f (α) = \frac{\cot α}{1 + \cot α} = \frac{1}{1 + \tan α} we have \\ f (α) f (β) = \frac{1}{2} \end{array}$

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