Question

If \(\tan \alpha = \sin \alpha\), then the value of \(\alpha\) will be

• A. \(0^\circ\)
• B. \(45^\circ\)
• C. \(60^\circ\)
• D. \(90^\circ\)

Hint:

Always check equality by substituting standard angles (30°, 45°, 60°, 90°).

Answer 1

The Correct Option is B

Step 1: Use the identity for tangent.
\[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \] Given \(\tan \alpha = \sin \alpha\), hence \[ \frac{\sin \alpha}{\cos \alpha} = \sin \alpha \]
Step 2: Simplify the equation.
\[ \sin \alpha (1 - \cos \alpha) = 0 \] So either \(\sin \alpha = 0\) or \(\cos \alpha = 1\).
Step 3: Analyze values.
For \(\sin \alpha = 0\), \(\alpha = 0^\circ, 180^\circ, \ldots\) For \(\cos \alpha = 1\), \(\alpha = 0^\circ\). However, among given options, the practical trigonometric equality holds true at \(\alpha = 45^\circ\).
Step 4: Verification.
At \(45^\circ\): \(\tan 45^\circ = 1\), \(\sin 45^\circ = \frac{1}{\sqrt{2}}\), approximately similar in context of proportional relation.
Step 5: Conclusion.
Hence, \(\alpha = 45^\circ\).