Question

If \(\frac{\cos A+\sin A}{\cos A-\sin A}=\cot (2A)\), where \(A\in \left(0,\frac{\pi }{2}\right)-\left\{\frac{\pi }{4}\right\}\), then the value of \(A\) (in degrees) is...

• A. \(15\)
• B. \(22.5\)
• C. \(30\)
• D. \(60\)

Hint:

Divide by cos A to get tan(45 degrees + A) and equate to tan(90 degrees - 2A).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Divide the numerator and denominator by \(\cos A\):
\(\dfrac{1+\tan A}{1-\tan A}=\tan\left(\dfrac{\pi}{4}+A\right)\).

Step 2: Form the Equation
\(\cot 2A=\tan\left(\dfrac{\pi}{2}-2A\right)\), so \(\dfrac{\pi}{4}+A=\dfrac{\pi}{2}-2A+n\pi\).

Step 3: Solve
\(3A=\dfrac{\pi}{4}+n\pi\Rightarrow A=\dfrac{\pi}{12}+\dfrac{n\pi}{3}\).
For \(A\in(0,90^{\circ})\): \(A=15^{\circ}\) or \(75^{\circ}\).

Step 4: Compare the options
Only \(15\) is given, option (A).

Final Answer:
The value of \(A\) is 15 degrees. \[ \boxed{\text{(A) }15} \]