Using trigonometric identities: Class 10 practice
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Trigonometric ratios
- Standard values
The idea behind using trigonometric identities
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. Use them to simplify expressions and to find all ratios from one.
Step-by-step method
- Identify which identity removes a square.
- Substitute and simplify.
- Check with a convenient angle like 30° or 60°.
Worked example 1 (Basic)
Simplify 3·(cosec²θ − 1) sin²θ.
- Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec²
- = 3cos²θ
- Check at θ = 30°: both sides ≈ 2.2500
- Answer: 3cos²θ
Worked example 2 (Standard)
Simplify 5·(cosec²θ − 1) sin²θ.
- Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec²
- = 5cos²θ
- Check at θ = 45°: both sides ≈ 2.5000
- Answer: 5cos²θ
Worked example 3 (Challenge)
Simplify 6·(1 + tan²θ)(1 − sinθ)(1 + sinθ).
- Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec²
- = 6
- Check at θ = 30°: both sides ≈ 6.0000
- Answer: 6
Common mistakes and misconceptions
- Writing sin²θ − cos²θ = 1
- Dropping the square
Practice questions
- (Basic) Simplify 3·(1 + tan²θ)(1 − sinθ)(1 + sinθ).
- (Basic) Simplify 4·(1 + tan²θ)(1 − sinθ)(1 + sinθ).
- (Standard) Simplify 6·(1 − cos²θ) sec²θ.
- (Standard) Simplify 2·sin⁴θ − cos⁴θ + cos²θ.
- (Challenge) Simplify 5·sin⁴θ − cos⁴θ + cos²θ.
- (Challenge) Simplify 3·(sec²θ − 1) cot²θ.
Answer key with full working
- Q1: Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec² → = 3 → Check at θ = 60°: both sides ≈ 3.0000 → Answer: 3
- Q2: Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec² → = 4 → Check at θ = 60°: both sides ≈ 4.0000 → Answer: 4
- Q3: Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec² → = 6tan²θ → Check at θ = 60°: both sides ≈ 18.0000 → Answer: 6tan²θ
- Q4: Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec² → = 2sin²θ → Check at θ = 45°: both sides ≈ 1.0000 → Answer: 2sin²θ
- Q5: Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec² → = 5sin²θ → Check at θ = 45°: both sides ≈ 2.5000 → Answer: 5sin²θ
- Q6: Replace using sin² + cos² = 1 / 1 + tan² = sec² / 1 + cot² = cosec² → = 3 → Check at θ = 45°: both sides ≈ 3.0000 → Answer: 3
Syllabus note
This skill is listed under "Trigonometry" for Class 10 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is using trigonometric identities in?
"Trigonometry", Class 10 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.