if sin theta = x/y then sec theta + cot theta =?

Given that \( \sin \theta = \frac{x}{y} \), we need to express \( \sec \theta + \cot \theta \) in terms of \( x \) and \( y \).

### Step 1: Use the Pythagorean Identity
Since \( \sin \theta = \frac{x}{y} \), we know:
\[
\cos^2 \theta + \sin^2 \theta = 1
\]
This implies:
\[
\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left( \frac{x}{y} \right)^2} = \sqrt{\frac{y^2 - x^2}{y^2}} = \frac{\sqrt{y^2 - x^2}}{y}
\]

### Step 2: Express \( \sec \theta \)
The secant function is the reciprocal of cosine, so:
\[
\sec \theta = \frac{1}{\cos \theta} = \frac{y}{\sqrt{y^2 - x^2}}
\]

### Step 3: Express \( \cot \theta \)
The cotangent function is the ratio of cosine to sine, so:
\[
\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{\sqrt{y^2 - x^2}}{y}}{\frac{x}{y}} = \frac{\sqrt{y^2 - x^2}}{x}
\]

### Step 4: Add \( \sec \theta \) and \( \cot \theta \)
Now, combine the expressions for \( \sec \theta \) and \( \cot \theta \):
\[
\sec \theta + \cot \theta = \frac{y}{\sqrt{y^2 - x^2}} + \frac{\sqrt{y^2 - x^2}}{x}
\]

This is the simplified expression for \( \sec \theta + \cot \theta \) in terms of \( x \) and \( y \).

Would you like further details or have any questions?

### Follow-up questions:
1. How does the Pythagorean identity relate to trigonometric functions?
2. Can you derive the relationship between secant and cosine?
3. Why is the cotangent function the ratio of cosine to sine?
4. Can this expression be simplified further for specific values of \( x \) and \( y \)?
5. How do trigonometric identities help in solving complex expressions?

### Tip:
Understanding the reciprocal identities (secant, cosecant, and cotangent) helps simplify many trigonometric problems quickly.

Explore how to solve the trigonometric expression sec(theta) + cot(theta) when sin(theta) = x/y. This problem involves key concepts such as the Pythagorean identity and reciprocal trigonometric functions, providing a step-by-step solution. Ideal for students in Grades 10-12.

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