Question

The value of \( (\csc A + \cot A)(1 - \cos A) \) will be:

• A. $\cos A$
• B. $\tan A$
• C. $\sec A$
• D. $\sin A$

Hint:

Always convert trigonometric terms into sine and cosine to simplify easily. Use \( 1 - \cos^2A = \sin^2A \) wherever applicable.

Answer 1

The Correct Option is D

Step 1: Write in terms of sine and cosine.
\[ (\csc A + \cot A)(1 - \cos A) = \left(\frac{1}{\sin A} + \frac{\cos A}{\sin A}\right)(1 - \cos A) \] \[ = \frac{(1 + \cos A)(1 - \cos A)}{\sin A} \]
Step 2: Simplify the numerator using an identity.
\[ (1 + \cos A)(1 - \cos A) = 1 - \cos^2 A = \sin^2 A \]
Step 3: Substitute back.
\[ (\csc A + \cot A)(1 - \cos A) = \frac{\sin^2 A}{\sin A} = \sin A \]
Step 4: Conclusion.
Hence, the value of \( (\csc A + \cot A)(1 - \cos A) \) is \( \boxed{\sin A} \).