Question

Which one of the following equations has no solution?

• A. $\csc \theta -\sec \theta =\csc \theta \cdot \sec \theta$
• B. $\csc \theta \cdot \sec \theta =1$
• C. $\cos \theta +\sin \theta =\sqrt{2}$
• D. $\sqrt{3}\sin \theta -\cos \theta =2$

Answer 1

The correct option is B $\csc \theta \cdot \sec \theta =1$

(a) $\csc \theta -\sec \theta =\csc \theta \cdot \sec \theta$
$\Rightarrow \frac{\cos \theta -\sin \theta }{\cos \theta \sin \theta }=\frac{1}{\cos \theta \sin \theta }$
$\Rightarrow \cos \theta =1+\sin \theta$
At $\theta =0$ above equation satisfies.
(b) $\csc \theta \cdot \sec \theta =1$
$\Rightarrow \sin \theta \cos \theta =1$
$\Rightarrow 2\sin \theta \cos \theta =2$
$\Rightarrow \sin 2\theta =2$
As we know $\sin \theta$ is not greater than $1$.
Therefore, the above equation has no solution exist.