Consider the following equations :
1 . `(cos 75^(@))/( sin 15 ^(@)) + (sin 12^(@))/( cos 78^(@)) - (cosec 18 ^(@))/( sec 72^(@)) = 1 `
2 . `(tan 50^(@) + sec 50^(@))/( cot 40^(@) + cosec 40^(@)) + cos 40^(@) cosec 50^(@) = +2 `
3. `(sin 80^(@))/( cos 10^(@)) - sin 59^(@) sec 31^(@) = 0 `
Which of these statements given below is correct

Options

A. 1 only is correct
B. 3 only is correct
C. All 1, 2 and 3 are correct
D. 2 and 3 are correct

Answer

To solve the given equations step by step, we will analyze each equation separately and verify their correctness.

### Equation 1:
\[ \frac{\cos 75^\circ}{\sin 15^\circ} + \frac{\sin 12^\circ}{\cos 78^\circ} - \frac{\csc 18^\circ}{\sec 72^\circ} = 1 \]

**Step 1: Simplify each term using trigonometric identities.**
- We know that \(\sin 15^\circ = \cos 75^\circ\) (since \(\sin(90^\circ - x) = \cos x\)).
- Also, \(\cos 78^\circ = \sin 12^\circ\) (again using the complementary angle identity).

Thus, we can rewrite the equation as:
\[ \frac{\cos 75^\circ}{\cos 75^\circ} + \frac{\sin 12^\circ}{\sin 12^\circ} - \frac{\csc 18^\circ}{\sec 72^\circ} = 1 \]

**Step 2: Evaluate the simplified terms.**
- The first term simplifies to \(1\).
- The second term also simplifies to \(1\).
- Now, we need to simplify \(\frac{\csc 18^\circ}{\sec 72^\circ}\):
\[ \csc 18^\circ = \frac{1}{\sin 18^\circ}, \quad \sec 72^\circ = \frac{1}{\cos 72^\circ} \]
Thus,
\[ \frac{\csc 18^\circ}{\sec 72^\circ} = \frac{\cos 72^\circ}{\sin 18^\circ} \]

Using the identity \(\sin 18^\circ = \cos 72^\circ\), we find:
\[ \frac{\cos 72^\circ}{\sin 18^\circ} = 1 \]

**Step 3: Substitute back into the equation.**
\[ 1 + 1 - 1 = 1 \]
Thus, the left-hand side equals the right-hand side, confirming that Equation 1 is correct.

### Equation 2:
\[ \frac{\tan 50^\circ + \sec 50^\circ}{\cot 40^\circ + \csc 40^\circ} + \cos 40^\circ \csc 50^\circ = 2 \]

**Step 1: Rewrite using identities.**
- \(\tan 50^\circ = \frac{1}{\cot 40^\circ}\) and \(\sec 50^\circ = \frac{1}{\cos 50^\circ}\).
- \(\cot 40^\circ = \frac{1}{\tan 40^\circ}\) and \(\csc 40^\circ = \frac{1}{\sin 40^\circ}\).

Thus, we can rewrite the equation as:
\[ \frac{\frac{1}{\cot 40^\circ} + \frac{1}{\cos 50^\circ}}{\cot 40^\circ + \csc 40^\circ} + \cos 40^\circ \csc 50^\circ = 2 \]

**Step 2: Simplify the left-hand side.**
Using the identity \(\cot 40^\circ + \csc 40^\circ = \frac{\sin 40^\circ + \cos 40^\circ}{\sin 40^\circ \cos 40^\circ}\), we can simplify further.

After simplification, we find:
\[ 1 + 1 = 2 \]
Thus, Equation 2 is also correct.

### Equation 3:
\[ \frac{\sin 80^\circ}{\cos 10^\circ} - \sin 59^\circ \sec 31^\circ = 0 \]

**Step 1: Simplify using identities.**
- We know \(\sin 80^\circ = \cos 10^\circ\).
- Also, \(\sec 31^\circ = \frac{1}{\cos 31^\circ}\).

Thus, we can rewrite the equation as:
\[ \frac{\cos 10^\circ}{\cos 10^\circ} - \sin 59^\circ \cdot \frac{1}{\cos 31^\circ} = 0 \]

**Step 2: Evaluate the terms.**
This simplifies to:
\[ 1 - \frac{\sin 59^\circ}{\cos 31^\circ} = 0 \]

Using the identity \(\sin 59^\circ = \cos 31^\circ\), we find:
\[ 1 - 1 = 0 \]
Thus, Equation 3 is also correct.

### Conclusion:
All three equations are correct. Therefore, the correct statements are:
1. Equation 1 is correct.
2. Equation 2 is correct.
3. Equation 3 is correct.