Question 1

Simplify: (1 - cos²θ) / sin θ

Accepted Answer

Step 1: Apply the Pythagorean identity sin²θ + cos²θ = 1, which gives 1 - cos²θ = sin²θ. Step 2: Substitute: sin²θ / sin θ. Step 3: Simplify by cancelling one sin θ: sin θ. Therefore, the answer is sin θ.

Question 2

Simplify: sec²θ - tan²θ

Accepted Answer

Step 1: Recall the trigonometric identity sec²θ - tan²θ = 1. Step 2: This is a direct application of the identity, which is derived from the Pythagorean identity sin²θ + cos²θ = 1 by dividing both sides by cos²θ. Step 3: Therefore, sec²θ - tan²θ = 1.

Question 3

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

Accepted Answer

Step 1: Apply the identity sin²θ + cos²θ = 1. Step 2: The expression becomes 1/(sin θ cos θ). Step 3: With sin θ = 3/5 and cos θ = 4/5: 1/(3/5 × 4/5) = 1/(12/25) = 25/12.

Question 4

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

Accepted Answer

Step 1: Use the identity tan θ = sin θ / cos θ, so sin θ / cos θ = 5/12. Step 2: Also use sin²θ + cos²θ = 1. Step 3: From tan θ = 5/12, we can construct a right triangle with opposite = 5 and adjacent = 12. Step 4: Hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13. Step 5: sin θ = opposite/hypotenuse = 5/13. Therefore, the answer is 5/13.

Question 5

Simplify: sin⁴θ - cos⁴θ + 2cos²θ

Accepted Answer

Step 1: Recognize sin⁴θ - cos⁴θ as a difference of squares: (sin²θ - cos²θ)(sin²θ + cos²θ). Step 2: Since sin²θ + cos²θ = 1, this becomes (sin²θ - cos²θ)(1) = sin²θ - cos²θ. Step 3: Now add 2cos²θ: sin²θ - cos²θ + 2cos²θ = sin²θ + cos²θ. Step 4: Apply the Pythagorean identity: sin²θ + cos²θ = 1.

Question 6

Simplify: (sec²θ - 1) / (sec²θ) × csc²θ

Accepted Answer

Step 1: Use the identity sec²θ - 1 = tan²θ. So the expression becomes (tan²θ / sec²θ) × csc²θ. Step 2: tan²θ = sin²θ / cos²θ and sec²θ = 1 / cos²θ. So tan²θ / sec²θ = (sin²θ / cos²θ) / (1 / cos²θ) = sin²θ. Step 3: Multiply by csc²θ = 1 / sin²θ. So sin²θ × (1 / sin²θ) = 1.

Question 7

Simplify: (1 - sin²θ) sec²θ + sin²θ

Accepted Answer

Step 1: Use the identity 1 - sin²θ = cos²θ. So the expression becomes cos²θ × sec²θ + sin²θ. Step 2: Recall that sec θ = 1/cos θ, so sec²θ = 1/cos²θ. Step 3: cos²θ × (1/cos²θ) = 1. Step 4: Add sin²θ: 1 + sin²θ.

Question 8

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 + 2tan θ cot θ + cot² θ = 16. Step 3: Use tan θ cot θ = 1: tan² θ + 2(1) + cot² θ = 16. Step 4: Simplify: tan² θ + cot² θ + 2 = 16. Step 5: Therefore tan² θ + cot² θ = 14.

Question 9

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

Accepted Answer

Step 1: Factor numerator: sin⁴ θ − cos⁴ θ = (sin² θ + cos² θ)(sin² θ − cos² θ). Step 2: Divide by denominator: [(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. Step 4: Rewrite using sin² θ + cos² θ = 1, so sin² θ = 1 − cos² θ: 1 + 2cos² θ = sin² θ + 3cos² θ. Alternatively: 1 + 2cos² θ = 2 − sin² θ.

Question 10

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

Accepted Answer

Step 1: Let sin θ − cos θ = 1/2. 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 = 3/4, so sin θ cos θ = 3/8. Step 5: Use the factorization sin³ θ − cos³ θ = (sin θ − cos θ)(sin² θ + sin θ cos θ + cos² θ). Step 6: Substitute: sin³ θ − cos³ θ = (1/2)[1 + 3/8] = (1/2)(11/8) = 11/16.

SBI Clerk Trig Identities

Trig Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Trig Identities — Practice Questions

60-Second Revision — Trig Identities