`sec^(2)theta + "cosec"^(2)theta` is equal to which of the following?

Options

A. `sectheta tan theta`
B. `sectheta "cosec"theta`
C. `sec^(2)theta "cosec"^(2)theta`
D. `sin^(4)theta + cos^(4)theta`

Answer

To solve the problem `sec²θ + cosec²θ`, we will follow these steps:

### Step 1: Write the definitions of secant and cosecant
We know that:
- \( \sec \theta = \frac{1}{\cos \theta} \)
- \( \csc \theta = \frac{1}{\sin \theta} \)

### Step 2: Substitute the definitions into the equation
Thus, we can rewrite \( sec²θ \) and \( cosec²θ \) as:
\[ sec²θ + cosec²θ = \frac{1}{\cos²θ} + \frac{1}{\sin²θ} \]

### Step 3: Find a common denominator
To combine these fractions, we need a common denominator, which is \( \cos²θ \sin²θ \):
\[ \frac{1}{\cos²θ} + \frac{1}{\sin²θ} = \frac{\sin²θ}{\cos²θ \sin²θ} + \frac{\cos²θ}{\cos²θ \sin²θ} \]

### Step 4: Combine the fractions
Now, we can combine the fractions:
\[ \frac{\sin²θ + \cos²θ}{\cos²θ \sin²θ} \]

### Step 5: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin²θ + \cos²θ = 1 \]
So we can substitute this into our equation:
\[ \frac{1}{\cos²θ \sin²θ} \]

### Step 6: Rewrite the expression in terms of secant and cosecant
This can be rewritten as:
\[ \frac{1}{\cos²θ} \cdot \frac{1}{\sin²θ} = \sec²θ \cdot \csc²θ \]

### Conclusion
Thus, we have:
\[ sec²θ + cosec²θ = \sec²θ \cdot \csc²θ \]

### Final Answer
The expression \( sec²θ + cosec²θ \) is equal to \( \sec²θ \cdot \csc²θ \).

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