Question 1

If cos θ = 5/13, what is the value of sin²θ?

Accepted Answer

Step 1: Use the identity sin²θ + cos²θ = 1, so sin²θ = 1 − cos²θ. Step 2: cos²θ = (5/13)² = 25/169. Step 3: sin²θ = 1 − 25/169 = (169 − 25)/169 = 144/169. Therefore, sin²θ = 144/169.

Question 2

If tan θ = 5/12, find sin θ (where θ is acute).

Accepted Answer

Step 1: Recognize tan θ = opposite/adjacent = 5/12. Step 2: Use the Pythagorean theorem to find the hypotenuse: h² = 5² + 12² = 25 + 144 = 169, so h = 13. Step 3: sin θ = opposite/hypotenuse = 5/13.

Question 3

Simplify: (1 - sin²θ) / cos θ

Accepted Answer

Step 1: Use the identity 1 - sin²θ = cos²θ. Step 2: Substitute: cos²θ / cos θ. Step 3: Simplify: cos²θ / cos θ = cos θ.

Question 4

If tan θ = 5/12, find the value of sec²θ - tan²θ.

Accepted Answer

Step 1: Use the identity sec²θ - tan²θ = 1 (a standard trigonometric identity). Step 2: This identity holds for all values of θ where tan θ is defined. Step 3: Therefore, sec²θ - tan²θ = 1, regardless of the value of tan θ.

Question 5

Simplify: (sin²θ + cos²θ) / sin²θ

Accepted Answer

Step 1: Use the identity sin²θ + cos²θ = 1. Step 2: Substitute into the numerator: 1 / sin²θ. Step 3: This equals cosec²θ.

Question 6

If tan θ = 5/12, find sin θ (where θ is acute).

Accepted Answer

Step 1: Recognize that tan θ = opposite/adjacent = 5/12. Step 2: Use the Pythagorean theorem to find the hypotenuse: h² = 5² + 12² = 25 + 144 = 169, so h = 13. Step 3: sin θ = opposite/hypotenuse = 5/13.

Question 7

Simplify: (1 - cos²θ) / sin θ

Accepted Answer

Step 1: Use the Pythagorean identity sin²θ + cos²θ = 1, which gives 1 - cos²θ = sin²θ. Step 2: Substitute: sin²θ / sin θ. Step 3: Simplify by canceling one sin θ: sin θ.

Question 8

Simplify: (sin²θ + cos²θ) / sin θ · cos θ

Accepted Answer

Step 1: Apply the Pythagorean identity sin²θ + cos²θ = 1. Step 2: Substitute: 1 / (sin θ · cos θ). Step 3: This is the simplified form. Therefore, answer is 1/(sin θ cos θ).

Question 9

Simplify: tan θ · cot θ + sin²θ

Accepted Answer

Step 1: Recall that tan θ · cot θ = 1 (since cot θ = 1/tan θ). Step 2: The expression becomes 1 + sin²θ. Step 3: This cannot be simplified further without additional information about θ.

Question 10

Simplify: (sin²θ + cos²θ) / sin θ × cos θ

Accepted Answer

Step 1: Use the identity sin²θ + cos²θ = 1. Step 2: Substitute into the expression: 1 / (sin θ × cos θ). Step 3: This equals 1/(sin θ cos θ), which cannot be simplified further without specific values. However, recognizing that the numerator equals 1 is the key simplification.

Question 11

Simplify: sin θ × csc θ + cos θ × sec θ

Accepted Answer

Step 1: Recall that csc θ = 1/sin θ and sec θ = 1/cos θ. Step 2: sin θ × (1/sin θ) + cos θ × (1/cos θ). Step 3: Simplify each term: 1 + 1 = 2.

Question 12

Simplify: sin²θ + sin²θ · cot²θ

Accepted Answer

Step 1: Factor out sin²θ: sin²θ(1 + cot²θ). Step 2: Apply the identity 1 + cot²θ = csc²θ. Step 3: Substitute: sin²θ · csc²θ. Step 4: Since csc θ = 1/sin θ, we have csc²θ = 1/sin²θ. Step 5: sin²θ · (1/sin²θ) = 1.

Question 13

Simplify: (sin⁴ A - cos⁴ A) / (sin² A - cos² A).

Accepted Answer

Step 1: Numerator = sin⁴ A - cos⁴ A = (sin² A - cos² A)(sin² A + cos² A) [difference of squares]. Step 2: Since sin² A + cos² A = 1, numerator = sin² A - cos² A. Step 3: Denominator = sin² A - cos² A. Step 4: Therefore, (sin² A - cos² A)/(sin² A - cos² A) = 1.

Question 14

Simplify: (sin⁴ A - cos⁴ A) / (sin² A - cos² A)

Accepted Answer

Step 1: Recognize that sin⁴ A - cos⁴ A is a difference of squares: (sin² A)² - (cos² A)² = (sin² A - cos² A)(sin² A + cos² A). Step 2: Substitute into the original expression: [(sin² A - cos² A)(sin² A + cos² A)] / (sin² A - cos² A). Step 3: Cancel (sin² A - cos² A) from numerator and denominator. Step 4: Use the Pythagorean identity sin² A + cos² A = 1. Therefore, the answer is 1.

Question 15

If tan θ + cot θ = 4, find the value of tan² θ + cot² θ.

Accepted Answer

Step 1: Given tan θ + cot θ = 4. Square both sides: (tan θ + cot θ)² = 16. Step 2: Expand: tan² θ + 2 tan θ cot θ + cot² θ = 16. Step 3: Since tan θ cot θ = 1 (reciprocal identity), we have: tan² θ + 2(1) + cot² θ = 16. Step 4: Therefore, tan² θ + cot² θ = 16 - 2 = 14.

Question 16

Prove that (1 - sin² θ) sec² θ = 1. What is the value of (1 - sin² θ) sec² θ when sin θ = 4/5?

Accepted Answer

Step 1: Simplify the left side using the Pythagorean identity 1 - sin² θ = cos² θ. Step 2: Substitute: cos² θ · sec² θ. Step 3: Since sec θ = 1/cos θ, we have sec² θ = 1/cos² θ. Step 4: Therefore, cos² θ · (1/cos² θ) = 1. Step 5: The expression equals 1 for all values of θ, including when sin θ = 4/5. The answer is 1.

Question 17

If tan θ + cot θ = 4, find the value of tan² θ + cot² θ.

Accepted Answer

Step 1: Let tan θ + cot θ = 4. Square both sides: (tan θ + cot θ)² = 16. Step 2: Expand: tan² θ + 2 tan θ cot θ + cot² θ = 16. Step 3: Since tan θ cot θ = 1 (reciprocal identity), we have tan² θ + 2(1) + cot² θ = 16. Step 4: Therefore, tan² θ + cot² θ = 16 - 2 = 14.

Question 18

Simplify: (tan² θ − sin² θ) / (tan² θ · sin² θ)

Accepted Answer

Step 1: Rewrite tan² θ as sin² θ / cos² θ. Step 2: Substitute into the numerator: (sin² θ / cos² θ) − sin² θ = sin² θ (1/cos² θ − 1) = sin² θ · (1 − cos² θ) / cos² θ = sin² θ · sin² θ / cos² θ = sin⁴ θ / cos² θ. Step 3: Substitute into the denominator: (sin² θ / cos² θ) · sin² θ = sin⁴ θ / cos² θ. Step 4: Divide numerator by denominator: (sin⁴ θ / cos² θ) ÷ (sin⁴ θ / cos² θ) = 1.

Question 19

If sin θ − cos θ = 1/2, then find the value of sin θ cos θ.

Accepted Answer

Step 1: Square both sides of sin θ − cos θ = 1/2. We get (sin θ − cos θ)² = 1/4. Step 2: Expand: sin² θ − 2 sin θ cos θ + cos² θ = 1/4. Step 3: Use the identity sin² θ + cos² θ = 1, so 1 − 2 sin θ cos θ = 1/4. Step 4: Solve for sin θ cos θ: −2 sin θ cos θ = 1/4 − 1 = −3/4. Therefore, sin θ cos θ = 3/8.

Question 20

If tan θ = 3/4 and θ is in the first quadrant, find the value of (5 sin θ + 4 cos θ) / (5 sin θ − 4 cos θ).

Accepted Answer

Step 1: From tan θ = 3/4, construct a right triangle with opposite = 3 and adjacent = 4. Step 2: Find hypotenuse: √(3² + 4²) = √25 = 5. Step 3: In the first quadrant, sin θ = 3/5 and cos θ = 4/5. Step 4: Substitute into the expression: (5 · 3/5 + 4 · 4/5) / (5 · 3/5 − 4 · 4/5) = (3 + 16/5) / (3 − 16/5) = (15/5 + 16/5) / (15/5 − 16/5) = (31/5) / (−1/5) = 31/5 · (−5/1) = −31.

Question 21

If sin θ + cos θ = √2, find the value of sin θ cos θ.

Accepted Answer

Step 1: Given sin θ + cos θ = √2. Square both sides: (sin θ + cos θ)² = 2. Step 2: Expand: sin² θ + 2 sin θ cos θ + cos² θ = 2. Step 3: Using sin² θ + cos² θ = 1: 1 + 2 sin θ cos θ = 2. Step 4: Therefore, 2 sin θ cos θ = 1, so sin θ cos θ = 1/2.

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