Question

If \[ \cos\theta-\sin\theta=\sqrt{5}\sin\theta, \] then \[ \cos\theta+4\sin\theta \] is equal to:

• A. \(5\cos\theta\)
• B. \(\sqrt{5}\sin\theta\)
• C. \(5\sin\theta\)
• D. \(\sqrt{5}\cos\theta\)

Hint:

When a trigonometric equation gives a relation between \(\sin\theta\) and \(\cos\theta\), express one in terms of the other and substitute in the required expression.

Answer 1

The Correct Option is D

Step 1: Start with the given equation.
Given, \[ \cos\theta-\sin\theta=\sqrt{5}\sin\theta \] Move \(-\sin\theta\) to the right side: \[ \cos\theta=\sqrt{5}\sin\theta+\sin\theta \] \[ \cos\theta=(\sqrt{5}+1)\sin\theta \]

Step 2: Express \(\sin\theta\) in terms of \(\cos\theta\).
From \[ \cos\theta=(\sqrt{5}+1)\sin\theta, \] we get \[ \sin\theta=\frac{\cos\theta}{\sqrt{5}+1} \]

Step 3: Evaluate \(\cos\theta+4\sin\theta\).
Now, \[ \cos\theta+4\sin\theta = \cos\theta+4\left(\frac{\cos\theta}{\sqrt{5}+1}\right) \] \[ = \cos\theta\left(1+\frac{4}{\sqrt{5}+1}\right) \] \[ = \cos\theta\left(\frac{\sqrt{5}+1+4}{\sqrt{5}+1}\right) \] \[ = \cos\theta\left(\frac{\sqrt{5}+5}{\sqrt{5}+1}\right) \] Now, \[ \sqrt{5}+5=\sqrt{5}(\sqrt{5}+1) \] Therefore, \[ \frac{\sqrt{5}+5}{\sqrt{5}+1}=\sqrt{5} \] So, \[ \cos\theta+4\sin\theta=\sqrt{5}\cos\theta \]

Step 4: Final conclusion.
Hence, \[ \boxed{\sqrt{5}\cos\theta} \]