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If $\frac{\sin^{3}\theta-\cos^{3}\theta}{\sin\theta-\cos\theta}-\frac{\cos\theta}{\sqrt{1+\cot^{2}\theta}}-\tan\theta \:\cot\theta=-1$ , then
Option: 1
$\theta\: \in \: \left ( 0,\frac{\pi}{2} \right )$

Option: 2
$\theta\:\in \:\left ( \frac{\pi}{2},\pi \right )$

Option: 3
$\theta\:\in \:\left (\pi, \frac{3\pi}{2} \right )$

Option: 4
$\theta\:\in \:\left ( \frac{3\pi}{2},2\pi \right )$

Answers (1)

best_answer

Trigonometric Identities -

$\sin ^{2}\Theta + \cos ^{2}\Theta = 1$

$1 + \tan ^{2}\Theta = \sec ^{2}\Theta$

$1 + \cot ^{2}\Theta = cosec ^{2}\Theta$

- wherein

They are true for all real values of $\Theta$

Since $\sin\theta-\cos\theta\neq 0$

$\tan\theta\neq 1$

and also to defined $\tan\theta,\theta\neq \frac{\pi}{2},\frac{3\pi}{2}$

$\therefore \theta\neq \frac{\pi}{4},\frac{5\pi}{4},\frac{\pi}{2},\frac{3\pi}{2}$

Now

$\sin ^{2}\theta+\cos ^{2}\theta+\sin\theta\:\cos\theta-\left | \sin\theta \right |\cos\theta-2\tan\theta\cot\theta=-1$

$\Rightarrow 1+\cos\theta(\sin\theta-\left | \sin\theta \right |)-2=-1$

$\Rightarrow \cos\theta(\sin\theta-\left | \sin\theta \right |)=0$

$\therefore \theta\:\varepsilon \:\left ( 0.\pi \right )-\left \{ \frac{\pi}{4},\frac{\pi}{2} \right \}$

Posted by

Irshad Anwar

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