Question

Simplify the following expression: \[ (\sec A - \cos A)(\tan A - \cot A) \] The simplified form is:

• A. \( \sin A (1 - \tan^2 A) \)
• B. \( -\sin A (1 - \tan^2 A) \)
• C. \( \cos A (1 + \cot^2 A) \)
• D. \( -\cos A (1 + \cot^2 A) \)
• E. \( 1 - \tan^2 A \)

Hint:

When simplifying trigonometric expressions, converting to sine and cosine can help simplify terms. Look for common identities such as \( 1 - \cos^2 A = \sin^2 A \) and factor where possible.

Answer 1

The Correct Option is B

We are tasked with simplifying the expression: \[ (\sec A - \cos A)(\tan A - \cot A). \] Step 1: First, express \( \sec A \) and \( \cot A \) in terms of \( \sin A \) and \( \cos A \):
- \( \sec A = \frac{1}{\cos A} \),
- \( \cot A = \frac{\cos A}{\sin A} \).
Substitute these into the given expression: \[ \left( \frac{1}{\cos A} - \cos A \right) \left( \tan A - \frac{\cos A}{\sin A} \right). \] Step 2: Simplify the first part \( \frac{1}{\cos A} - \cos A \). We get a common denominator: \[ \frac{1}{\cos A} - \cos A = \frac{1 - \cos^2 A}{\cos A}. \] Using the identity \( 1 - \cos^2 A = \sin^2 A \), this becomes: \[ \frac{\sin^2 A}{\cos A}. \] Step 3: Now, simplify the second part \( \tan A - \frac{\cos A}{\sin A} \): \[ \tan A = \frac{\sin A}{\cos A}, \quad \frac{\cos A}{\sin A} = \cot A. \] Thus, we have: \[ \frac{\sin A}{\cos A} - \frac{\cos A}{\sin A}. \] To combine these terms, find a common denominator: \[ \frac{\sin^2 A - \cos^2 A}{\sin A \cos A}. \] This can be written as: \[ \frac{-(\cos^2 A - \sin^2 A)}{\sin A \cos A} = - \frac{\cos 2A}{\sin A \cos A}. \] Step 4: Now multiply the two parts: \[ \frac{\sin^2 A}{\cos A} \times \left( -\frac{\cos^2 A - \sin^2 A}{\sin A \cos A} \right) = - \frac{\sin A (1 - \tan^2 A)}{1}. \] Thus, the simplified expression is: \[ - \sin A (1 - \tan^2 A). \]
Therefore, the correct answer is option (B).