ICSE Class 10 Maths Trig Identities Solutions Selina
ICSE Class 10 Maths Chapter 21 trigonometrical identities
ICSE Class 10 Maths Chapter 21, Trigonometrical Identities, teaches you to prove that two trigonometric expressions are equal by using standard identities and algebraic simplification. This page is for the Selina Concise Mathematics Class 10 chapter and gives a formula sheet, proof method, original worked examples, exam presentation notes and common mistake corrections.
In an identity proof, you do not find the value of the angle. You start from one side, simplify it step by step, and show that it becomes the other side. The safest method is to begin with the more complicated side and reduce the number of different ratios.
Concept snapshot: reduce every expression to fewer ratios
Most proof questions become easier when you reduce the expression to \sin A and \cos A. If a denominator contains 1-\sin A, 1+\sin A, 1-\cos A, or 1+\cos A, use the conjugate. If powers appear, look for a^2-b^2, \sin^2 A+\cos^2 A=1, 1+\tan^2 A=\sec^2 A, or 1+\cot^2 A=\cosec^2 A.
Formula reference for Concise Mathematics Selina Solutions Class 10 ICSE Chapter 21 Trigonometrical Identities
| Identity type | Formula | Use in proof |
|---|---|---|
| Basic identity | \sin^2 A+\cos^2 A=1 | Use when both sine-square and cosine-square terms occur. |
| Tangent-square identity | 1+\tan^2 A=\sec^2 A | Use for \sec^2 A-1 or \tan^2 A. |
| Cotangent-square identity | 1+\cot^2 A=\cosec^2 A | Use for \cosec^2 A-1 or \cot^2 A. |
| Quotient identities | \tan A=\frac{\sin A}{\cos A}, \cot A=\frac{\cos A}{\sin A} | Use when a common denominator is needed. |
| Reciprocal identities | \sec A=\frac{1}{\cos A}, \cosec A=\frac{1}{\sin A} | Use when secant or cosecant blocks the simplification. |
Method to prove trigonometrical identities in Maths
- Choose the side: Start from the side with more terms, brackets, powers or mixed ratios.
- Convert when stuck: Write \tan A, \cot A, \sec A, and \cosec A in terms of \sin A and \cos A.
- Use the right identity: Do not apply every formula blindly. Apply the one that reduces the expression.
- Use conjugates: For denominators such as 1-\sin A and 1+\cos A, multiply by the matching conjugate.
- End clearly: Write \text{LHS}=\text{RHS}, hence proved.
Edition note: Exercise labels and the number of sub-parts may vary between printings of the textbook. The proof methods below remain the same for the Chapter 21 identity types.
Worked examples from Chapter 21 Trigonometrical Identities
Example 1: Simplify \sin^4 \theta-\cos^4 \theta
Step 1: Use a^2-b^2=(a+b)(a-b).
\sin^4 \theta-\cos^4 \theta=(\sin^2 \theta+\cos^2 \theta)(\sin^2 \theta-\cos^2 \theta)
Step 2: Use \sin^2 \theta+\cos^2 \theta=1.
(\sin^2 \theta+\cos^2 \theta)(\sin^2 \theta-\cos^2 \theta)=\sin^2 \theta-\cos^2 \theta
Step 3: Replace \cos^2 \theta with 1-\sin^2 \theta.
\sin^2 \theta-\cos^2 \theta=\sin^2 \theta-(1-\sin^2 \theta)=2\sin^2 \theta-1
Final answer: \sin^4 \theta-\cos^4 \theta=2\sin^2 \theta-1.
Example 2: Prove \dfrac{1}{1-\sin A}=\sec^2 A(1+\sin A)
Step 1: Start from the left-hand side.
\text{LHS}=\frac{1}{1-\sin A}
Step 2: Multiply numerator and denominator by the conjugate 1+\sin A.
\frac{1}{1-\sin A}\times\frac{1+\sin A}{1+\sin A}=\frac{1+\sin A}{1-\sin^2 A}
Step 3: Use 1-\sin^2 A=\cos^2 A.
\frac{1+\sin A}{1-\sin^2 A}=\frac{1+\sin A}{\cos^2 A}=\sec^2 A(1+\sin A)
Hence proved: \dfrac{1}{1-\sin A}=\sec^2 A(1+\sin A).
Example 3: Prove \dfrac{\sec A-1}{\sec A+1}=\dfrac{1-\cos A}{1+\cos A}
Step 1: Begin with the left-hand side and write \sec A=\frac{1}{\cos A}.
\text{LHS}=\frac{\sec A-1}{\sec A+1}=\frac{\frac{1}{\cos A}-1}{\frac{1}{\cos A}+1}
Step 2: Simplify both compound fractions.
\frac{\frac{1-\cos A}{\cos A}}{\frac{1+\cos A}{\cos A}}=\frac{1-\cos A}{1+\cos A}
Hence proved: \dfrac{\sec A-1}{\sec A+1}=\dfrac{1-\cos A}{1+\cos A}.
Example 4: Prove \sec A(1-\sin A)(\sec A+\tan A)=1
Step 1: Convert \sec A and \tan A into sine and cosine forms.
\text{LHS}=\frac{1}{\cos A}(1-\sin A)\left(\frac{1}{\cos A}+\frac{\sin A}{\cos A}\right)
Step 2: Combine the last bracket.
\frac{1}{\cos A}(1-\sin A)\left(\frac{1+\sin A}{\cos A}\right)=\frac{(1-\sin A)(1+\sin A)}{\cos^2 A}
Step 3: Use the difference of squares and the basic identity.
\frac{(1-\sin A)(1+\sin A)}{\cos^2 A}=\frac{1-\sin^2 A}{\cos^2 A}=\frac{\cos^2 A}{\cos^2 A}=1
Hence proved: \sec A(1-\sin A)(\sec A+\tan A)=1.
Examiner’s mindset: proof presentation
A proof earns credit through a correct chain of reasoning. Show the starting side, write the identity or substitution used, simplify the algebra, and finish with a clear equality. Do not jump from the first line to the final answer. Also avoid changing both sides at once unless every step is reversible and clearly shown.
Common mistakes students make
- Wrong cancellation: Cancel common factors, not separate terms in a sum such as \sin A+\cos A.
- Mixing identities: 1+\tan^2 A=\sec^2 A, while 1+\cot^2 A=\cosec^2 A.
- Using the wrong conjugate: The conjugate of 1-\sin A is 1+\sin A, not 1-\cos A.
- Stopping early: A proof is complete only when the simplified side exactly matches the other side.
Quick answer index
| Item | Final result |
|---|---|
| Example 1 | \sin^4 \theta-\cos^4 \theta=2\sin^2 \theta-1 |
| Example 2 | \dfrac{1}{1-\sin A}=\sec^2 A(1+\sin A) |
| Example 3 | \dfrac{\sec A-1}{\sec A+1}=\dfrac{1-\cos A}{1+\cos A} |
| Example 4 | \sec A(1-\sin A)(\sec A+\tan A)=1 |
Related ICSE Class 10 Maths revision links
For connected revision, use Selina Concise Maths Class 10 solutions, heights and distances solutions, quadratic equations solutions, and ratio and proportion solutions.
For official syllabus and specimen-paper references, use the CISCE website: cisceboard.org.
Frequently Asked Questions
How should I start a trigonometrical identity proof in ICSE Class 10 Maths?
Start from the more complicated side, convert reciprocal and quotient ratios into sine and cosine when needed, apply a basic identity, and finish only when the expression matches the other side exactly.
Which identities are most important in Selina Chapter 21 Trigonometrical Identities?
The main identities are \sin^2 A+\cos^2 A=1, 1+\tan^2 A=\sec^2 A, and 1+\cot^2 A=\cosec^2 A. Reciprocal and quotient identities are also needed in most proofs.
Can I prove a trigonometric identity by working on both sides together?
It is safer to simplify one side at a time and then show that it equals the other side. This keeps the proof clear and reduces presentation errors.
Why is rationalisation used in trigonometrical identities?
Rationalisation helps when the denominator has 1-\sin A, 1+\sin A, 1-\cos A, or 1+\cos A. Multiplying by the conjugate creates 1-\sin^2 A or 1-\cos^2 A, which can be simplified.
Is Chapter 21 enough for ICSE trigonometry revision?
No. Chapter 21 builds identity-proof skills, but ICSE Class 10 Maths trigonometry revision should also include ratio values, complementary angles and heights and distances.