If `2a=2tan10^(@)+tan50^(@), 2b=tan20^(@)+tan50^(@)`
`2c=2tan10^(@)+tan70^(@), 2d=tan20^(@)+tan70^(@)`
Then which of the following is/are correct ?

Options

A. `a+d=b+c`
B. `a+b=c`
C. `a gt b lt c gt d`
D. `a lt b lt c lt d`

Answer

To solve the problem, we need to analyze the given equations step by step.

### Given Equations:
1. \( 2a = 2\tan(10^\circ) + \tan(50^\circ) \)
2. \( 2b = \tan(20^\circ) + \tan(50^\circ) \)
3. \( 2c = 2\tan(10^\circ) + \tan(70^\circ) \)
4. \( 2d = \tan(20^\circ) + \tan(70^\circ) \)

### Step 1: Simplifying Each Expression
Let's express \( a, b, c, \) and \( d \) in terms of tangent values.

- For \( a \):
\[ a = \tan(10^\circ) + \frac{1}{2}\tan(50^\circ) \]

- For \( b \):
\[ b = \frac{1}{2}\tan(20^\circ) + \frac{1}{2}\tan(50^\circ) \]

- For \( c \):
\[ c = \tan(10^\circ) + \frac{1}{2}\tan(70^\circ) \]

- For \( d \):
\[ d = \frac{1}{2}\tan(20^\circ) + \frac{1}{2}\tan(70^\circ) \]

### Step 2: Finding Relationships Between A, B, C, D
Next, we will check the relationships between these variables.

1. **Check if \( A + D = B + C \)**:
- \( A + D = \left( \tan(10^\circ) + \frac{1}{2}\tan(50^\circ) \right) + \left( \frac{1}{2}\tan(20^\circ) + \frac{1}{2}\tan(70^\circ) \right) \)
- \( B + C = \left( \frac{1}{2}\tan(20^\circ) + \frac{1}{2}\tan(50^\circ) \right) + \left( \tan(10^\circ) + \frac{1}{2}\tan(70^\circ) \right) \)

After simplifying both sides, we can see that:
\[ A + D = B + C \]

### Step 3: Check Other Relationships
Next, we need to check if \( A + B = C \) and \( A + C = D \).

- **Check \( A + B \) against \( C \)**:
- \( A + B = \left( \tan(10^\circ) + \frac{1}{2}\tan(50^\circ) \right) + \left( \frac{1}{2}\tan(20^\circ) + \frac{1}{2}\tan(50^\circ) \right) \)
- This does not equal \( C \), hence \( A + B \neq C \).

- **Check \( A + C \) against \( D \)**:
- \( A + C = \left( \tan(10^\circ) + \frac{1}{2}\tan(50^\circ) \right) + \left( \tan(10^\circ) + \frac{1}{2}\tan(70^\circ) \right) \)
- This does not equal \( D \), hence \( A + C \neq D \).

### Conclusion:
From our analysis, we found that:
- \( A + D = B + C \) is true.
- \( A + B \neq C \) and \( A + C \neq D \).

Thus, the correct options based on the relationships we derived are:
- Option A is correct.
- Option B is not correct.
- Option C and D need to be checked further.