Question 1

Simplify: (1 - cos²θ) / sin θ

Accepted Answer

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

Question 2

Simplify: sec²θ - tan²θ

Accepted Answer

Step 1: Recall the trigonometric identity sec²θ - tan²θ = 1. Step 2: This is a standard identity derived from the Pythagorean identity sin²θ + cos²θ = 1 by dividing both sides by cos²θ. Step 3: The simplified form is 1.

Question 3

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: This means in a right triangle, opposite = 5k and adjacent = 12k for some k. Step 3: Hypotenuse = √(5k)² + (12k)² = √(25k² + 144k²) = √(169k²) = 13k. Step 4: sin θ = opposite/hypotenuse = 5k/13k = 5/13.

Question 4

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

Accepted Answer

Step 1: Simplify (1 - sin² θ) sec² θ using the identity 1 - sin² θ = cos² θ. Step 2: cos² θ × sec² θ = cos² θ × (1/cos² θ) = 1. Step 3: The identity is always 1, regardless of the value of sin θ. Step 4: When sin θ = 4/5, the left side still equals 1.

Question 5

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) = (sin² A - cos² A)(1). Step 2: Denominator: sin² A - cos² A. Step 3: Simplify: (sin² A - cos² A)(1) / (sin² A - cos² A) = 1.

Question 6

If tan θ + cot θ = 4, then tan²θ + cot²θ equals:

Accepted Answer

Step 1: Given tan θ + cot θ = 4. Step 2: Square both sides: (tan θ + cot θ)² = 16. Step 3: Expand: tan²θ + 2·tan θ·cot θ + cot²θ = 16. Step 4: Since tan θ·cot θ = 1, we have tan²θ + 2 + cot²θ = 16. Step 5: Therefore, tan²θ + cot²θ = 14.

Question 7

If sin θ - cos θ = 1/2, then sin θ · cos θ equals:

Accepted Answer

Step 1: Given sin θ - cos θ = 1/2. Step 2: Square both sides: (sin θ - cos θ)² = 1/4. Step 3: Expand: sin²θ - 2·sin θ·cos θ + cos²θ = 1/4. Step 4: Use sin²θ + cos²θ = 1: 1 - 2·sin θ·cos θ = 1/4. Step 5: Solve: 2·sin θ·cos θ = 3/4, so sin θ·cos θ = 3/8.

Question 8

If cos²θ - sin²θ = 1/3, then cos⁴θ - sin⁴θ equals:

Accepted Answer

Step 1: Given cos²θ - sin²θ = 1/3. Step 2: Use the factorization a⁴ - b⁴ = (a² - b²)(a² + b²). Step 3: cos⁴θ - sin⁴θ = (cos²θ - sin²θ)(cos²θ + sin²θ). Step 4: Substitute cos²θ + sin²θ = 1 and cos²θ - sin²θ = 1/3: cos⁴θ - sin⁴θ = (1/3)(1) = 1/3.

Question 9

If sec θ + tan θ = 5, then sec θ - tan θ equals:

Accepted Answer

Step 1: Given sec θ + tan θ = 5. Step 2: Use the identity sec²θ - tan²θ = 1. Step 3: Factor: (sec θ - tan θ)(sec θ + tan θ) = 1. Step 4: Substitute sec θ + tan θ = 5: (sec θ - tan θ)(5) = 1. Step 5: Solve: sec θ - tan θ = 1/5.

SBI PO Trig Identities

Trig Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Trig Identities — Practice Questions

60-Second Revision — Trig Identities