Pythagorean Identities in Trigonometry Worksheets - Free Printable

To solve the given problems, we need to verify the Pythagorean identities. Let's go through each problem step by step.

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Problem 1:

Verify that:
\[
\cos \theta = \sec \theta \cdot \cos^2 \theta
\]

#### Solution:
1. Recall the definition of the secant function:
\[
\sec \theta = \frac{1}{\cos \theta}
\]

2. Substitute this definition into the right-hand side of the equation:
\[
\sec \theta \cdot \cos^2 \theta = \left( \frac{1}{\cos \theta} \right) \cdot \cos^2 \theta
\]

3. Simplify the expression:
\[
\left( \frac{1}{\cos \theta} \right) \cdot \cos^2 \theta = \frac{\cos^2 \theta}{\cos \theta} = \cos \theta
\]

4. The left-hand side is already \(\cos \theta\), so we have shown that:
\[
\cos \theta = \cos \theta
\]

Thus, the identity is verified.

\[
\boxed{\text{Verified}}
\]

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Problem 2:

Verify that:
\[
\frac{1 - \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 + \cos \theta}
\]

#### Solution:
1. Start with the left-hand side:
\[
\frac{1 - \cos \theta}{\sin \theta}
\]

2. Multiply the numerator and the denominator by the conjugate of the numerator, \(1 + \cos \theta\):
\[
\frac{1 - \cos \theta}{\sin \theta} \cdot \frac{1 + \cos \theta}{1 + \cos \theta} = \frac{(1 - \cos \theta)(1 + \cos \theta)}{\sin \theta (1 + \cos \theta)}
\]

3. Simplify the numerator using the difference of squares:
\[
(1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta
\]

4. Recall the Pythagorean identity:
\[
1 - \cos^2 \theta = \sin^2 \theta
\]

5. Substitute this into the expression:
\[
\frac{1 - \cos^2 \theta}{\sin \theta (1 + \cos \theta)} = \frac{\sin^2 \theta}{\sin \theta (1 + \cos \theta)}
\]

6. Simplify by canceling \(\sin \theta\) in the numerator and the denominator:
\[
\frac{\sin^2 \theta}{\sin \theta (1 + \cos \theta)} = \frac{\sin \theta}{1 + \cos \theta}
\]

7. This matches the right-hand side of the original equation:
\[
\frac{\sin \theta}{1 + \cos \theta}
\]

Thus, the identity is verified.

\[
\boxed{\text{Verified}}
\]

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Problem 3:

Verify that:
\[
\cot^2 x + \csc^2 x = \csc^4 x - \csc^2 x
\]

#### Solution:
1. Recall the definitions of the cotangent and cosecant functions:
\[
\cot x = \frac{\cos x}{\sin x}, \quad \csc x = \frac{1}{\sin x}
\]

2. Use the Pythagorean identity for cotangent and cosecant:
\[
\cot^2 x + 1 = \csc^2 x
\]

3. Rearrange this identity to express \(\cot^2 x\):
\[
\cot^2 x = \csc^2 x - 1
\]

4. Substitute \(\cot^2 x = \csc^2 x - 1\) into the left-hand side of the given equation:
\[
\cot^2 x + \csc^2 x = (\csc^2 x - 1) + \csc^2 x
\]

5. Simplify the expression:
\[
(\csc^2 x - 1) + \csc^2 x = 2\csc^2 x - 1
\]

6. Now, consider the right-hand side of the equation:
\[
\csc^4 x - \csc^2 x
\]

7. Factor out \(\csc^2 x\) from the right-hand side:
\[
\csc^4 x - \csc^2 x = \csc^2 x (\csc^2 x - 1)
\]

8. Recall the Pythagorean identity again:
\[
\csc^2 x - 1 = \cot^2 x
\]

9. Substitute \(\csc^2 x - 1 = \cot^2 x\) into the factored expression:
\[
\csc^2 x (\csc^2 x - 1) = \csc^2 x \cdot \cot^2 x
\]

10. However, we notice a discrepancy here. Let's re-evaluate the right-hand side directly:
\[
\csc^4 x - \csc^2 x = \csc^2 x (\csc^2 x - 1)
\]

11. Using \(\csc^2 x - 1 = \cot^2 x\):
\[
\csc^2 x (\csc^2 x - 1) = \csc^2 x \cdot \cot^2 x
\]

12. This does not directly match the left-hand side. Let's recheck the problem statement. It appears there might be a typo or an error in the problem as stated. However, based on standard identities, the correct verification should align with the Pythagorean relationships.

Given the standard identities, the correct form should be:
\[
\cot^2 x + 1 = \csc^2 x
\]

Thus, the identity as stated may need correction. Assuming the problem is correctly stated, the verification process shows a potential discrepancy.

\[
\boxed{\text{Verification pending correction}}
\]

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Final Answers:

1. \(\boxed{\text{Verified}}\)
2. \(\boxed{\text{Verified}}\)
3. \(\boxed{\text{Verification pending correction}}\)