Question 1

If cos θ = 7/25, find tan θ (where θ is acute).

Accepted Answer

Step 1: From cos θ = 7/25, we have adjacent = 7 and hypotenuse = 25. Step 2: Use the Pythagorean theorem: opposite² = 25² - 7² = 625 - 49 = 576, so opposite = 24. Step 3: tan θ = opposite/adjacent = 24/7.

Question 2

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

Accepted Answer

Step 1: Recall 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: Apply the Pythagorean identity: 1 - sin²θ = cos²θ. Step 2: Substitute: cos²θ / cos θ. Step 3: Simplify by cancelling one factor of cos θ: cos θ.

Question 4

If tan θ = 5/12, find sin θ (assuming θ is in the first quadrant).

Accepted Answer

Step 1: Recall that tan θ = sin θ / cos θ = 5/12. Step 2: Use the identity sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. Step 3: If tan θ = 5/12, then in a right triangle, opposite = 5 and adjacent = 12. Step 4: Hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13. Step 5: sin θ = opposite/hypotenuse = 5/13.

Question 5

If tan θ = 5/12, find sin θ (assuming θ is in the first quadrant).

Accepted Answer

Step 1: Use the identity tan θ = sin θ / cos θ = 5/12. Step 2: In a right triangle, if opposite = 5 and adjacent = 12, then hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13. Step 3: sin θ = opposite/hypotenuse = 5/13. Therefore, answer is 5/13.

Question 6

Simplify: tan θ · cot θ + sec²θ - tan²θ

Accepted Answer

Step 1: Recognize that tan θ · cot θ = 1 (reciprocal identity). Step 2: Recognize that sec²θ - tan²θ = 1 (Pythagorean identity). Step 3: Combine: 1 + 1 = 2.

Question 7

Evaluate: sin²60° + cos²60°

Accepted Answer

Step 1: Apply the Pythagorean identity sin²θ + cos²θ = 1 for any angle θ. Step 2: This identity is universal and holds for θ = 60° as well. Step 3: sin²60° + cos²60° = 1. Therefore, answer is 1.

Question 8

Simplify: sin²30° + cos²30°

Accepted Answer

Step 1: Apply the Pythagorean identity sin²θ + cos²θ = 1 for any angle θ. Step 2: This identity holds for θ = 30° as well. Step 3: sin²30° + cos²30° = 1. Therefore, answer is 1.

Question 9

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

Accepted Answer

Step 1: Apply the Pythagorean identity sin²θ + cos²θ = 1. Step 2: Substitute into the expression: 1 / (sin θ · cos θ). Step 3: This equals sec θ · csc θ, but the simplest form is 1/(sin θ cos θ).

Question 10

Prove that (1 − sin² θ) sec² θ = 1. Which of the following is the correct first step in the proof?

Accepted Answer

Step 1: Start with the left side: (1 − sin² θ) sec² θ. Step 2: Use the Pythagorean identity 1 − sin² θ = cos² θ. Step 3: Substitute: cos² θ · sec² θ. Step 4: Since sec θ = 1/cos θ, we have sec² θ = 1/cos² θ. Step 5: Therefore: cos² θ · (1/cos² θ) = 1. The correct first step is to replace (1 − sin² θ) with cos² θ.

Question 11

Simplify: (sin⁴ α − cos⁴ α) / (sin² α − cos² α)

Accepted Answer

Step 1: Recognize that sin⁴ α − cos⁴ α is a difference of squares: (sin² α)² − (cos² α)² = (sin² α − cos² α)(sin² α + cos² α). Step 2: Divide by (sin² α − cos² α): [(sin² α − cos² α)(sin² α + cos² α)] / (sin² α − cos² α) = sin² α + cos² α. Step 3: Apply the Pythagorean identity: sin² α + cos² α = 1.

Question 12

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, we have: tan² θ + 2(1) + cot² θ = 16. Step 4: Therefore, tan² θ + cot² θ = 16 − 2 = 14.

Question 13

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

Accepted Answer

Step 1: Numerator is a difference of squares: sin⁴ α - cos⁴ α = (sin² α - cos² α)(sin² α + cos² α). Step 2: Using sin² α + cos² α = 1, numerator becomes (sin² α - cos² α) × 1 = sin² α - cos² α. Step 3: Divide by denominator: (sin² α - cos² α)/(sin² α - cos² α) = 1.

Question 14

Prove that (1 - sin² θ) sec² θ = 1. What is the value of (1 - sin² θ) sec² θ when sin θ = 0.6?

Accepted Answer

Step 1: Simplify the identity: (1 - sin² θ) sec² θ = cos² θ × (1/cos² θ) = 1. This is true for all values of θ where cos θ ≠ 0. Step 2: The expression always equals 1, regardless of the value of sin θ. Step 3: When sin θ = 0.6, the expression still equals 1.

Question 15

Simplify: (sin θ / (1 - cos θ)) - (sin θ / (1 + cos θ))

Accepted Answer

Step 1: Find common denominator: [(sin θ)(1 + cos θ) - (sin θ)(1 - cos θ)] / [(1 - cos θ)(1 + cos θ)]. Step 2: Numerator = sin θ(1 + cos θ) - sin θ(1 - cos θ) = sin θ + sin θ cos θ - sin θ + sin θ cos θ = 2 sin θ cos θ. Step 3: Denominator = 1 - cos² θ = sin² θ. Step 4: Result = 2 sin θ cos θ / sin² θ = 2 cos θ / sin θ = 2 cot θ.

Question 16

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: Apply the identity sin² A + cos² A = 1. Step 3: Numerator becomes 1 × (sin² A - cos² A) = (sin² A - cos² A). Step 4: Divide by denominator (sin² A - cos² A) to get 1.

Question 17

Prove that (1 - sin² θ) sec² θ = 1. Which of the following is the correct sequence of steps?

Accepted Answer

Step 1: Start with LHS = (1 - sin² θ) sec² θ. Step 2: Apply the Pythagorean identity: 1 - sin² θ = cos² θ. Step 3: Substitute: cos² θ × sec² θ. Step 4: Recall that sec θ = 1/cos θ, so sec² θ = 1/cos² θ. Step 5: Simplify: cos² θ × (1/cos² θ) = 1 = RHS. Therefore, the identity is proven.

Question 18

If sin θ − cos θ = 1/2, then find 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 − 2sin θ cos θ + cos² θ = 1/4. Step 3: Use sin² θ + cos² θ = 1, so 1 − 2sin θ cos θ = 1/4. Step 4: Solve: −2sin θ cos θ = 1/4 − 1 = −3/4. Step 5: sin θ cos θ = 3/8.

Question 19

If sin θ - cos θ = 1/2, then find sin θ cos θ.

Accepted Answer

Step 1: Square both sides: (sin θ - cos θ)² = 1/4. Step 2: Expand: sin² θ - 2sin θ cos θ + cos² θ = 1/4. Step 3: Use sin² θ + cos² θ = 1: 1 - 2sin θ cos θ = 1/4. Step 4: Solve: -2sin θ cos θ = 1/4 - 1 = -3/4. Step 5: Therefore, sin θ cos θ = 3/8.

Question 20

If sec θ - tan θ = 2/5, then find sec θ + tan θ.

Accepted Answer

Step 1: Use the identity sec² θ - tan² θ = 1, which factors as (sec θ - tan θ)(sec θ + tan θ) = 1. Step 2: Substitute sec θ - tan θ = 2/5: (2/5)(sec θ + tan θ) = 1. Step 3: Solve: sec θ + tan θ = 1 ÷ (2/5) = 5/2.

RRB NTPC Trig Identities

Trig Identities— Rules & Concept

Key Points to Remember

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Trig Identities — Practice Questions

60-Second Revision — Trig Identities