Question

Which statements are true? Select all that apply.
(1 point)
Responses

1. tangent squared theta equals 1 minus secant squared theta
Image with alt text: tangent squared theta equals 1 minus secant squared theta

2. cosine theta equals Start Fraction cotangent theta over cosecant theta End Fraction
Image with alt text: cosine theta equals Start Fraction cotangent theta over cosecant theta End Fraction

3. Start Fraction 1 over secant squared theta End Fraction equals sine squared theta plus 1
Image with alt text: Start Fraction 1 over secant squared theta End Fraction equals sine squared theta plus 1

4. secant squared theta minus 1 equals Start Fraction 1 over cotangent squared theta End Fraction
Image with alt text: secant squared theta minus 1 equals Start Fraction 1 over cotangent squared theta End Fraction

5. sin θ csc θ = tan θ

Answer 1

Let's analyze each statement one by one to determine its truth.

1. **tangent squared theta equals 1 minus secant squared theta**
**False**. The correct identity is:
\[ \tan^2(\theta) + 1 = \sec^2(\theta) \]
So, \(\tan^2(\theta) = \sec^2(\theta) - 1\).

2. **cosine theta equals Start Fraction cotangent theta over cosecant theta End Fraction**
**True**. The cotangent can be defined as \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\) and cosecant as \( \csc(\theta) = \frac{1}{\sin(\theta)}\). Therefore:
\[ \frac{\cot(\theta)}{\csc(\theta)} = \frac{\frac{\cos(\theta)}{\sin(\theta)}}{\frac{1}{\sin(\theta)}} = \cos(\theta) \]

3. **Start Fraction 1 over secant squared theta End Fraction equals sine squared theta plus 1**
**False**. The correct identity is:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \] which means:
\[ \frac{1}{\sec^2(\theta)} = \frac{1}{1 + \tan^2(\theta)} \neq \sin^2(\theta) + 1\]

4. **secant squared theta minus 1 equals Start Fraction 1 over cotangent squared theta End Fraction**
**True**. This can be confirmed using the identity \(\sec^2(\theta) - 1 = \tan^2(\theta)\) and that \(\tan^2(\theta) = \frac{1}{\cot^2(\theta)}\):
\[ \sec^2(\theta) - 1 = \tan^2(\theta) = \frac{1}{\cot^2(\theta)} \]

5. **sin θ csc θ = tan θ**
**False**. The left side is equal to:
\[ \sin(\theta) \cdot \csc(\theta) = \sin(\theta) \cdot \frac{1}{\sin(\theta)} = 1 \]
while \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), which is not necessarily equal to 1.

### Conclusion:
The true statements are:
- Statement 2: **True**
- Statement 4: **True**

So the correct answers are 2 and 4.