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: Rearrange the given equation.
Given $\cos\theta - \sin\theta = \sqrt{5}\sin\theta$. Move $\sin\theta$ to the right: \[ \cos\theta = (\sqrt{5}+1)\sin\theta \]

Step 2: Find $\sin\theta$ in terms of $\cos\theta$.
\[ \sin\theta = \frac{\cos\theta}{\sqrt{5}+1} \] Rationalize: multiply numerator and denominator by $(\sqrt{5}-1)$: \[ \sin\theta = \frac{(\sqrt{5}-1)\cos\theta}{(\sqrt{5}+1)(\sqrt{5}-1)} = \frac{(\sqrt{5}-1)\cos\theta}{4} \]

Step 3: Compute $\cos\theta + 4\sin\theta$.
\[ \cos\theta + 4\sin\theta = \cos\theta + 4 \cdot \frac{(\sqrt{5}-1)\cos\theta}{4} = \cos\theta + (\sqrt{5}-1)\cos\theta = \sqrt{5}\cos\theta \]

Step 4: Verify using direct substitution.
From Step 1: $\sin\theta = \frac{\cos\theta}{\sqrt{5}+1}$. Then $4\sin\theta = \frac{4\cos\theta}{\sqrt{5}+1}$. Note $\frac{\sqrt{5}+5}{\sqrt{5}+1} = \sqrt{5}$. So $\cos\theta + 4\sin\theta = \cos\theta(1 + \frac{4}{\sqrt{5}+1}) = \cos\theta \cdot \frac{\sqrt{5}+5}{\sqrt{5}+1} = \sqrt{5}\cos\theta$.

Step 5: Confirm the answer matches an option.
Option 4 is $\sqrt{5}\cos\theta$. This matches exactly.

Step 6: State the answer.
\[ \boxed{\sqrt{5}\cos\theta} \]