Consider the following equations
(i) `cosec^2 x + sec^2 x = cosec^2 x sec^2 x`
(ii) `sec^2 x + tan^2 x = secx tan^2 x`
(iii) `cosec^2 x+ tan^2x = cot^2x + sec^2x`
Which of the above equations are correct ?

Options

A. (i) and (ii)
B. (ii) and (iii)
C. (i) and (iii)
D. (i), (ii) and (iii)

Answer

To determine which of the given equations are correct, we will analyze each equation step by step.

### Step 1: Analyze the First Equation
The first equation is:
\[ \csc^2 x + \sec^2 x = \csc^2 x \sec^2 x \]

**Left Hand Side (LHS)**:
\[ \csc^2 x + \sec^2 x = \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x} \]
To combine these fractions, we need a common denominator:
\[ = \frac{\cos^2 x + \sin^2 x}{\sin^2 x \cos^2 x} \]
Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\):
\[ = \frac{1}{\sin^2 x \cos^2 x} = \csc^2 x \sec^2 x \]

**Right Hand Side (RHS)**:
\[ \csc^2 x \sec^2 x = \frac{1}{\sin^2 x} \cdot \frac{1}{\cos^2 x} = \frac{1}{\sin^2 x \cos^2 x} \]

Since LHS = RHS, the first equation is **correct**.

### Step 2: Analyze the Second Equation
The second equation is:
\[ \sec^2 x + \tan^2 x = \sec x \tan^2 x \]

**Left Hand Side (LHS)**:
\[ \sec^2 x + \tan^2 x = \frac{1}{\cos^2 x} + \frac{\sin^2 x}{\cos^2 x} = \frac{1 + \sin^2 x}{\cos^2 x} \]
Using the identity \(1 + \tan^2 x = \sec^2 x\):
\[ = \sec^2 x \]

**Right Hand Side (RHS)**:
\[ \sec x \tan^2 x = \frac{1}{\cos x} \cdot \frac{\sin^2 x}{\cos^2 x} = \frac{\sin^2 x}{\cos^3 x} \]

Since LHS does not equal RHS, the second equation is **incorrect**.

### Step 3: Analyze the Third Equation
The third equation is:
\[ \csc^2 x + \tan^2 x = \cot^2 x + \sec^2 x \]

**Left Hand Side (LHS)**:
\[ \csc^2 x + \tan^2 x = \frac{1}{\sin^2 x} + \frac{\sin^2 x}{\cos^2 x} \]
Finding a common denominator:
\[ = \frac{\cos^2 x + \sin^4 x}{\sin^2 x \cos^2 x} \]

**Right Hand Side (RHS)**:
\[ \cot^2 x + \sec^2 x = \frac{\cos^2 x}{\sin^2 x} + \frac{1}{\cos^2 x} \]
Finding a common denominator:
\[ = \frac{\cos^4 x + \sin^2 x}{\sin^2 x \cos^2 x} \]

Since both sides simplify to the same expression, the third equation is **correct**.

### Conclusion
The correct equations are:
- First Equation: Correct
- Second Equation: Incorrect
- Third Equation: Correct