First slide

Trigonometric equations

Question

If $\sin (π \cot θ) = \cos (π \tan θ)$ then

Moderate

A

$\cot 2 θ = \frac{1}{4}, - \frac{3}{4}$
B

$\cot 2 θ = 4, \frac{4}{3}$
C

$\cot 2 θ = - \frac{3}{4}, - \frac{1}{4}$
D

none of these

Solution

We have,
$\begin{array}{l} \sin (π \cot θ) = \cos (π \tan θ) \\ \sin (π \cot θ) = \sin (\frac{π}{2} + π \tan θ) \\ \cos (π \tan θ) = \cos (\frac{3 π}{2} + π \cot θ) \end{array}$
$\begin{array}{l} \Rightarrow π \cot θ = \frac{π}{2} + π \tan θ or, π \tan θ = \frac{3 π}{2} + π \cot θ \\ \Rightarrow \cot θ - \tan θ = \frac{1}{2} or, \cot θ - \tan θ = - \frac{3}{2} \\ \Rightarrow \frac{1 - \tan^{2} θ}{2 \tan θ} = \frac{1}{4} or, \frac{1 - \tan^{2} θ}{2 \tan θ} = - \frac{3}{4} \\ \Rightarrow \cot 2 θ = \frac{1}{4} or, \cot 2 θ = - \frac{3}{4} \end{array}$

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