Question

If \[ 1-\cot 23^\circ=\frac{x}{1-\cot 22^\circ}, \] then \(x=\)

• A. \(1\)
• B. \(2\)
• C. \(\frac{1}{2}\)
• D. \(3\)

Hint:

When two angles add up to \(45^\circ\), expressions involving \((1-\cot A)(1-\cot B)\) often simplify using the tangent addition formula.

Answer 1

The Correct Option is B

Step 1: Rearrange the given equation.
Given, \[ 1-\cot 23^\circ=\frac{x}{1-\cot 22^\circ}. \] Multiplying both sides by \(1-\cot 22^\circ\), we get \[ x=(1-\cot 23^\circ)(1-\cot 22^\circ). \]

Step 2: Use the identity for cotangent.
We know that \[ \cot \theta=\frac{\cos\theta}{\sin\theta}. \] So, \[ 1-\cot\theta = 1-\frac{\cos\theta}{\sin\theta} = \frac{\sin\theta-\cos\theta}{\sin\theta}. \] Thus, \[ x= \frac{\sin23^\circ-\cos23^\circ}{\sin23^\circ} \cdot \frac{\sin22^\circ-\cos22^\circ}{\sin22^\circ}. \]

Step 3: Convert using complementary angles.
Since \[ \cos23^\circ=\sin67^\circ \] and \[ \cos22^\circ=\sin68^\circ, \] we can simplify using standard trigonometric transformations.
Also, \[ 23^\circ+22^\circ=45^\circ. \] This complementary relation helps simplify the product \[ (1-\cot23^\circ)(1-\cot22^\circ). \]

Step 4: Use the identity.
For angles \(A\) and \(B\) such that \[ A+B=45^\circ, \] we have \[ (1-\cot A)(1-\cot B)=2. \] Here, \[ A=23^\circ,\qquad B=22^\circ. \] Since \[ 23^\circ+22^\circ=45^\circ, \] we get \[ (1-\cot23^\circ)(1-\cot22^\circ)=2. \] Therefore, \[ x=2. \]

Step 5: Final conclusion.
Hence, \[ \boxed{2} \]