Question

If \( \csc(90^\circ + A) + x \cos A \cot(90^\circ + A) = \sin(90^\circ + A) \), then the value of

• A. \( \cot A \)
• B. \( \csc A \)
• C. \( \tan A \)
• D. \( \sin A \)

Hint:

For trigonometric equations, always use the standard identities and simplify the equation step by step to solve for the unknowns.

Answer 1

The Correct Option is C

We are given the equation: \[ \csc(90^\circ + A) + x \cos A \cot(90^\circ + A) = \sin(90^\circ + A) \] Using the following trigonometric identities: \[ \csc(90^\circ + A) = \sec A, \quad \cot(90^\circ + A) = -\tan A, \quad \sin(90^\circ + A) = \cos A \] Substituting these into the equation: \[ \sec A + x \cos A (-\tan A) = \cos A \] Simplifying: \[ \sec A - x \cos A \tan A = \cos A \] Now, \( \sec A = \frac{1}{\cos A} \), so the equation becomes: \[ \frac{1}{\cos A} - x \cos A \tan A = \cos A \] Rearranging the terms: \[ \frac{1}{\cos A} = \cos A + x \cos A \tan A \] Simplifying further, we find that the value of \( x \) is \( \tan A \). Hence, the correct value is \( \tan A \). Thus, the answer is \( \tan A \).