Question 1

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

Accepted Answer

Step 1: Recall cos θ = adjacent/hypotenuse = 7/25. Step 2: Use Pythagoras to find opposite: o² + 7² = 25², so o² = 625 - 49 = 576, thus o = 24. Step 3: tan θ = opposite/adjacent = 24/7. Therefore, the answer is 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 Pythagoras to find hypotenuse: h² = 5² + 12² = 25 + 144 = 169, so h = 13. Step 3: sin θ = opposite/hypotenuse = 5/13. Therefore, the answer is 5/13.

Question 3

Simplify: (sin²45° + cos²45°) × tan 60°

Accepted Answer

Step 1: Recall that sin²45° + cos²45° = 1 (Pythagorean identity). Step 2: Recall that tan 60° = √3. Step 3: Multiply: 1 × √3 = √3. Therefore, the answer is √3.

Question 4

Simplify: (1 - cos²θ) sec²θ

Accepted Answer

Step 1: Use the identity 1 - cos²θ = sin²θ. Step 2: The expression becomes sin²θ · sec²θ. Step 3: Since sec θ = 1/cos θ, we have sec²θ = 1/cos²θ. Step 4: sin²θ · (1/cos²θ) = sin²θ/cos²θ = tan²θ.

Question 5

Prove that (sec²θ - 1) cot²θ = 1. What is the value of (sec²θ - 1) cot²θ?

Accepted Answer

Step 1: Use the identity sec²θ - 1 = tan²θ. Step 2: The expression becomes tan²θ · cot²θ. Step 3: Since cot θ = 1/tan θ, we have cot²θ = 1/tan²θ. Step 4: tan²θ · (1/tan²θ) = 1.

Question 6

Simplify: (cos⁴θ - sin⁴θ)/(cos²θ - sin²θ)

Accepted Answer

Step 1: Factor the numerator using difference of squares: cos⁴θ - sin⁴θ = (cos²θ - sin²θ)(cos²θ + sin²θ). Step 2: The expression becomes [(cos²θ - sin²θ)(cos²θ + sin²θ)]/(cos²θ - sin²θ). Step 3: Cancel (cos²θ - sin²θ) from numerator and denominator. Step 4: Simplify to cos²θ + sin²θ = 1.

Question 7

Prove that (1 - sin² θ)(1 + tan² θ) = 1. What is the simplified form of the left side before equating to 1?

Accepted Answer

Step 1: Simplify 1 - sin² θ = cos² θ using the Pythagorean identity. Step 2: Simplify 1 + tan² θ = sec² θ using the identity 1 + tan² θ = sec² θ. Step 3: Multiply: cos² θ × sec² θ = cos² θ × (1/cos² θ) = 1. The simplified form before equating to 1 is cos² θ · sec² θ.

Question 8

If sec θ - tan θ = 2/5, find the value of 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 into the factored form: (2/5)(sec θ + tan θ) = 1. Step 3: Solve for sec θ + tan θ: sec θ + tan θ = 1 ÷ (2/5) = 5/2.

Question 9

Simplify: (sin⁴ θ - cos⁴ θ)/(sin² θ - cos² 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: Use the Pythagorean identity sin² θ + cos² θ = 1. So numerator = (sin² θ - cos² θ)(1) = sin² θ - cos² θ. Step 3: Divide numerator by denominator: (sin² θ - cos² θ)/(sin² θ - cos² θ) = 1.

SSC MTS Trig Identities

Trig Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Trig Identities — Practice Questions

60-Second Revision — Trig Identities