Question : If $\sin\theta+\cos\theta=\sqrt{2}\cos\theta$, then the value of $\cot\theta$ is:
Option 1: $\sqrt{2}+1$
Option 2: $\sqrt{2}-1$
Option 3: $\sqrt{3}-1$
Option 4: $\sqrt{3}+1$
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Correct Answer: $\sqrt{2}+1$
Solution :
$\sin\theta+\cos\theta=\sqrt{2}\cos\theta$
⇒ $\sin \theta = (\sqrt{2}-1)\cos\theta$
⇒ $\frac{\sin \theta}{(\sqrt{2}-1)}=\cos\theta$
⇒ $\frac{\sin \theta(\sqrt{2}+1)}{2-1}=\cos\theta$
⇒ $\frac{\sin \theta(\sqrt{2}+1)}{2-1}=\cos\theta$
⇒ $\frac{\cos \theta}{\sin \theta}=\sqrt{2}+1$
$\therefore\cot \theta=\sqrt{2}+1$
Hence, the correct answer is $\sqrt{2}+1$.
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