Question 1

Simplify: sin²θ + sin²θ · cot²θ

Accepted Answer

Step 1: Factor out sin²θ: sin²θ(1 + cot²θ). Step 2: Use 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. Therefore, answer is 1.

Question 2

Simplify: (sin θ · sec θ) / tan θ

Accepted Answer

Step 1: Rewrite using definitions: sec θ = 1/cos θ and tan θ = sin θ/cos θ. Step 2: Substitute: (sin θ · 1/cos θ) / (sin θ/cos θ). Step 3: Simplify the numerator: sin θ/cos θ. Step 4: Divide: (sin θ/cos θ) / (sin θ/cos θ) = 1. Therefore, answer is 1.

Question 3

If cos θ = 7/25 and θ is acute, find tan θ.

Accepted Answer

Step 1: cos θ = adjacent/hypotenuse = 7/25. Step 2: Use the Pythagorean theorem: opposite² + 7² = 25², so opposite² = 625 - 49 = 576, thus opposite = 24. Step 3: tan θ = opposite/adjacent = 24/7. Therefore, answer is 24/7.

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 = opposite/adjacent = 5/12. Step 2: Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13. Step 3: sin θ = opposite/hypotenuse = 5/13.

Question 5

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

Accepted Answer

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

Question 6

Simplify: (1 - cos²θ) / sin²θ

Accepted Answer

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

Question 7

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

Accepted Answer

Step 1: Recognize that the numerator is a difference of squares: sin⁴ α − cos⁴ α = (sin² α)² − (cos² α)² = (sin² α + cos² α)(sin² α − cos² α). Step 2: Apply the Pythagorean identity: sin² α + cos² α = 1. Step 3: The numerator becomes 1 × (sin² α − cos² α) = (sin² α − cos² α). Step 4: Divide by the denominator: (sin² α − cos² α)/(sin² α − cos² α) = 1.

Question 8

If sin θ − cos θ = 1/2, find the value of sin³ θ − cos³ θ.

Accepted Answer

Step 1: Square sin θ − cos θ = 1/2 to get sin² θ − 2sin θ cos θ + cos² θ = 1/4. Step 2: Use sin² θ + cos² θ = 1, so 1 − 2sin θ cos θ = 1/4, giving sin θ cos θ = 3/8. Step 3: Use the factorization a³ − b³ = (a − b)(a² + ab + b²). Step 4: sin³ θ − cos³ θ = (sin θ − cos θ)(sin² θ + sin θ cos θ + cos² θ) = (1/2)(1 + 3/8) = (1/2)(11/8) = 11/16.

Question 9

Simplify: (sec θ - cos θ) / (sec θ) and express in terms of sin θ.

Accepted Answer

Step 1: sec θ = 1/cos θ. Step 2: Numerator = 1/cos θ - cos θ = (1 - cos² θ)/cos θ = sin² θ / cos θ. Step 3: Divide by sec θ: (sin² θ / cos θ) ÷ (1/cos θ) = (sin² θ / cos θ) × cos θ = sin² θ.

Question 10

Simplify: (sin⁴ θ − cos⁴ θ)/(sin² θ − cos² θ) + 2cos² θ.

Accepted Answer

Step 1: Factor the numerator using difference of squares: sin⁴ θ − cos⁴ θ = (sin² θ − cos² θ)(sin² θ + cos² θ). Step 2: Simplify the fraction: (sin⁴ θ − cos⁴ θ)/(sin² θ − cos² θ) = (sin² θ − cos² θ)(sin² θ + cos² θ)/(sin² θ − cos² θ) = sin² θ + cos² θ = 1. Step 3: Add 2cos² θ: 1 + 2cos² θ.

Question 11

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. Step 2: (sin θ - cos θ)² = (1/2)² = 1/4. Step 3: sin² θ - 2sin θ cos θ + cos² θ = 1/4. Step 4: Using sin² θ + cos² θ = 1, we get 1 - 2sin θ cos θ = 1/4. Step 5: -2sin θ cos θ = 1/4 - 1 = -3/4. Step 6: sin θ cos θ = 3/8.

Question 12

If tan θ + cot θ = 4, then find tan² θ + cot² θ.

Accepted Answer

Step 1: Square both sides of tan θ + cot θ = 4. Step 2: (tan θ + cot θ)² = 16. Step 3: tan² θ + 2tan θ cot θ + cot² θ = 16. Step 4: Since tan θ · cot θ = 1, we have tan² θ + 2(1) + cot² θ = 16. Step 5: tan² θ + cot² θ = 16 - 2 = 14.

Question 13

Simplify: (tan² θ - sin² θ) / (tan² θ · sin² θ).

Accepted Answer

Step 1: Rewrite tan² θ = sin² θ / cos² θ. Step 2: 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: Denominator = (sin² θ / cos² θ) · sin² θ = sin⁴ θ / cos² θ. Step 4: (sin⁴ θ / cos² θ) ÷ (sin⁴ θ / cos² θ) = 1.

Question 14

If 3sin θ + 4cos θ = 5, then find sin θ · cos θ.

Accepted Answer

Step 1: By Cauchy-Schwarz inequality, (3sin θ + 4cos θ)² ≤ (3² + 4²)(sin² θ + cos² θ) = 25 × 1 = 25. Step 2: Equality holds when 3sin θ / 3 = 4cos θ / 4, i.e., sin θ = cos θ is false. Equality holds when sin θ / 3 = cos θ / 4 (proportionality). Step 3: Let sin θ = 3k, cos θ = 4k. Step 4: From sin² θ + cos² θ = 1: 9k² + 16k² = 1 → 25k² = 1 → k = 1/5. Step 5: sin θ = 3/5, cos θ = 4/5. Step 6: sin θ cos θ = 12/25.

SSC CHSL Trig Identities

Trig Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Trig Identities — Practice Questions

60-Second Revision — Trig Identities