If `2x` and `2y` are complementary angles and `tan (x+2y)=2`, then which of the following is (are) correct ?

Options

A. `sin (x+y)=1/2`
B. `tan(x-y)=1/7`
C. `cot x+cot y = 5`
D. `tan x tany =6`

Answer

To solve the problem step by step, we start with the given conditions:

1. **Understanding Complementary Angles**:
- We know that \(2x\) and \(2y\) are complementary angles. This means:
\[ 2x + 2y = 90^\circ \]
- Dividing the entire equation by 2 gives:
\[ x + y = 45^\circ \]

2. **Using the Given Equation**:
- We are given that:
\[ \tan(x + 2y) = 2 \]
- We can express \(x + 2y\) in terms of \(x\) and \(y\):
\[ x + 2y = x + (45^\circ - x) + y = 45^\circ + y \]
- Therefore, we can rewrite the equation as:
\[ \tan(45^\circ + y) = 2 \]

3. **Applying the Tangent Addition Formula**:
- The tangent addition formula states:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
- Here, \(A = 45^\circ\) and \(B = y\). Since \(\tan(45^\circ) = 1\), we have:
\[ \tan(45^\circ + y) = \frac{1 + \tan y}{1 - \tan y} \]
- Setting this equal to 2 gives:
\[ \frac{1 + \tan y}{1 - \tan y} = 2 \]

4. **Cross-Multiplying**:
- Cross-multiplying yields:
\[ 1 + \tan y = 2(1 - \tan y) \]
- Simplifying this leads to:
\[ 1 + \tan y = 2 - 2\tan y \]
- Rearranging gives:
\[ 3\tan y = 1 \implies \tan y = \frac{1}{3} \]

5. **Finding Cotangent of y**:
- Since \(\tan y = \frac{1}{3}\), we can find \(\cot y\):
\[ \cot y = \frac{1}{\tan y} = 3 \]

6. **Finding Cotangent of x**:
- Using the identity \(\tan(90^\circ - \theta) = \cot \theta\), we can find \(\cot x\):
\[ \cot x = \tan(45^\circ - y) = \frac{1 - \tan y}{1 + \tan y} = \frac{1 - \frac{1}{3}}{1 + \frac{1}{3}} = \frac{\frac{2}{3}}{\frac{4}{3}} = \frac{1}{2} \]

7. **Verifying Options**:
- Now we can check the options given in the problem:
- **Option A**: \(\sin(x + y) = \frac{1}{2}\) is false since \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\).
- **Option B**: \(\tan x - y = \frac{1}{7}\) can be verified using the values of \(\tan x\) and \(\tan y\).
- **Option C**: \(\cot x + \cot y = 5\) is true since \(3 + \frac{1}{2} = 5\).
- **Option D**: \(\tan x \tan y = 6\) is false since \(\tan x \tan y = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}\).

Thus, the correct options are B and C.