First slide

Trigonometric Identities

Question

$\frac{\sin^{3} θ - \cos^{3} θ}{\sin θ - \cos θ} - \frac{\cos θ}{\sqrt{1 + \cot^{2} θ}} - 2 \tan θcot θ = - 1$ if

Easy

A

$θ \in (0, \frac{π}{2})$
B

$θ \in (\frac{π}{2}, π)$
C

$θ \in (π, \frac{3 π}{2})$
D

$θ \in (\frac{3 π}{2}, 2 π)$

Solution

$(\sin^{2} θ + \cos^{2} θ + \sin θcos θ) - \frac{\cos θ}{| cosec θ |} - 2 = - 1$
$\begin{array}{l} or 1 + \sin θcos θ - | \sin θ | \cos θ - 1 = 0 \\ or \cos θ (\sin θ - | \sin θ |) = 0 \end{array}$
$\Rightarrow θ \in (0, \frac{π}{2}) or θ \in (\frac{π}{2}, π)$
But $θ \neq \frac{π}{4}$ (not in domain);
Therefore $θ \in (\frac{π}{2}, π)$

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