Chapter 8: Introduction to Trigonometry
Unique Previous Year Questions extracted from the CBSE Class 10 Standard Mathematics 2025 question paper sets. Repetitive questions have been removed while retaining the important question types and variations.
A
Multiple Choice Questions
1. Acute Angle – Sine
1 Mark
If \(\theta\) is an acute angle and
\(7+4\sin\theta=9\), then the value of \(\theta\) is:
Series 30/1/1 (Q4) & 30/1/3 (Q5)
2. Trigonometric Simplification
1 Mark
The value of
\[
\tan^2\theta-
\left(\frac{1}{\cos\theta}\times\sec\theta\right)
\]
is:
Series 30/1/1 (Q5)
3. Trigonometric Expression
1 Mark
The value of
\[
(\tan A\csc A)^2-(\sin A\sec A)^2
\]
is:
Series 30/1/3 (Q2)
4. Sine–Cosine Relation
1 Mark
If
\[
7\cos^2\theta+3\sin^2\theta=4,
\]
then the value of \(\theta\) is:
Series 30/2/1 (Q1)
5. Half Angle
1 Mark
If
\[
\sin(\alpha+\beta)=1,
\]
then the value of
\[
\sin\left(\frac{\alpha+\beta}{2}\right)
\]
is:
Series 30/2/2 (Q11)
6. Standard Angles
1 Mark
If
\[
\sin30^\circ\tan45^\circ
=
\frac{\sec60^\circ}{k},
\]
then the value of \(k\) is:
Series 30/2/2 (Q18)
7. Sine Equals Cosine
1 Mark
If
\[
\sin\theta=\cos\theta,
\qquad 0^\circ<\theta<90^\circ,
\]
then the value of
\[
\sec\theta\cdot\sin\theta
\]
is:
Series 30/2/3 (Q2)
8. Multiple Angle
1 Mark
If
\[
\tan3\theta=\sqrt3,
\]
then
\[
\frac{\theta}{2}
\]
equals:
Series 30/3/1 (Q1) & 30/3/3 (Q7)
9. Cotangent–Tangent Identity
1 Mark
\[
\cot\theta+\tan\theta
\]
equals:
Series 30/3/1 (Q9), 30/3/2 (Q11) & 30/3/3 (Q11)
10. Multiple Angle – Sine
1 Mark
If
\[
\sin4\theta=\frac{\sqrt3}{2},
\]
then
\[
\frac{\theta}{3}
\]
equals:
Series 30/3/2 (Q5)
11. Trigonometric Ratio Transformation
1 Mark
\[
\frac{\cos\theta}
{\sqrt{1-\cos^2\theta}}
\]
is equal to:
Series 30/3/2 (Q13)
12. Double Angle
1 Mark
\[
\tan2A=3\tan A
\]
is true, when the measure of \(\angle A\) is:
Series 30/4/1 (Q8)
13. Comparison of Trigonometric Ratios
1 Mark
Which of the following statements is true?
Series 30/4/1 (Q9) & 30/4/2 (Q11)
14. Trigonometric Identity
1 Mark
Which of the following is a trigonometric identity?
Series 30/4/2 (Q10)
15. Tangent of Double Angle
1 Mark
The value of
\[
\frac{2\tan60^\circ}{1-\tan^260^\circ}
\]
is same as the value of:
Series 30/5/2 (Q12)
16. Right Triangle – Secant
1 Mark
In a right triangle \(ABC\), right-angled at \(A\), if
\[
\sin B=\frac14,
\]
then the value of \(\sec B\) is:
Series 30/6/2 (Q2) & 30/6/3 (Q9)
17. Standard Angles – Algebraic Relation
1 Mark
If
\[
x=2\sin60^\circ\cos60^\circ
\]
and
\[
y=\sin^230^\circ-\cos^230^\circ
\]
and
\[
x^2=ky^2,
\]
then the value of \(k\) is:
Series 30/6/3 (Q3)
B
Very Short Answer Questions
18. Standard Angles – Algebraic Values
2 Marks
If
\[
x\cos60^\circ+y\cos0^\circ+\sin30^\circ-\cot45^\circ=5,
\]
then find the value of \(x+2y\).
Series 30/1/1 (Q21a) & 30/1/3 (Q22a)
19. Evaluation Using Standard Angles
2 Marks
Evaluate:
\[
\frac{\tan^260^\circ}
{\sin^260^\circ+\cos^230^\circ}.
\]
Series 30/1/3 (Q22b)
20. Given Tangent – Find Expression
2 Marks
If
\[
\tan A=\sqrt3,
\]
where \(A\) is an acute angle, then find:
\[
\frac{\sin^2A}{1+\cos^2A}.
\]
Series 30/2/2 (Q25) & 30/2/3 (Q24)
21. Standard Angles – Find \(k\)
2 Marks
If
\[
4k=\tan^260^\circ-2\csc^230^\circ-2\tan^230^\circ,
\]
then find the value of \(k\).
Series 30/3/1 (Q21)
22. Tangent and Cotangent
2 Marks
If
\[
\tan A+\cot A=6,
\]
then find:
\[
\tan^2A+\cot^2A-4.
\]
Series 30/3/3 (Q21)
23. Using Sine Difference Formula
2 Marks
It is given that
\[
\sin(A-B)=\sin A\cos B-\cos A\sin B.
\]
Use it to find the value of
\[
\sin15^\circ.
\]
Series 30/5/1 (Q23a)
24. Express Ratios in Terms of \(y\)
2 Marks
If
\[
\sin A=y,
\]
then express \(\cos A\) and \(\tan A\) in terms of \(y\).
Series 30/5/1 (Q23b)
C
Short Answer Questions
25. Prove the Trigonometric Identity
3 Marks
Prove that:
\[
\frac{\tan\theta}{1-\cot\theta}
+
\frac{\cot\theta}{1-\tan\theta}
=
1+\sec\theta\csc\theta
\]
Series 30/1/1 (Q27a)
26. Prove the Identity Using Sine and Cosine
3 Marks
Prove that:
\[
\frac{\sin A+\cos A}{\sin A-\cos A}
+
\frac{\sin A-\cos A}{\sin A+\cos A}
=
\frac{2}{2\sin^2A-1}
\]
Series 30/1/1 (Q27b)
27. Prove the Identity Using Reciprocal Ratios
3 Marks
Prove that:
\[
\left(1+\frac1{\tan^2\theta}\right)
\left(1+\frac1{\cot^2\theta}\right)
=
\frac1{\sin^2\theta-\sin^4\theta}
\]
Series 30/2/1 (Q27a)
28. Prove the Identity Involving Cosecant
3 Marks
Prove that:
\[
\sqrt{\frac{\csc\theta-1}{\csc\theta+1}}
+
\sqrt{\frac{\csc\theta+1}{\csc\theta-1}}
=
2\sec\theta
\]
Series 30/2/1 (Q27b)
29. Algebraic Form of Tangent and Sine
3 Marks
If
\[
\tan\theta+\sin\theta=m
\]
and
\[
\tan\theta-\sin\theta=n,
\]
then prove that:
\[
m^2-n^2=4\sqrt{mn}.
\]
Series 30/2/2 (Q26a)
30. Prove the Identity
3 Marks
Prove that:
\[
\frac{\cot A-1}{2-\sec^2A}
=
\frac{\cot A}{1+\tan A}
\]
Series 30/2/2 (Q26b)
31. Prove the Identity Involving Secant and Cosecant
3 Marks
Prove that:
\[
\sqrt{\sec^2\theta+\csc^2\theta}
=
\tan\theta+\cot\theta
\]
Series 30/2/3 (Q26a)
32. Cosecant–Cotangent Relation
3 Marks
If
\[
\csc\theta=x+\frac1{4x},
\]
prove that
\[
\csc\theta+\cot\theta=2x
\]
or
\[
\frac1{2x}.
\]
Series 30/2/3 (Q26b)
33. Prove the Identity Involving Secant
3 Marks
Prove that:
\[
\sqrt{\frac{\sec A-1}{\sec A+1}}
+
\sqrt{\frac{\sec A+1}{\sec A-1}}
=
2\csc A
\]
Series 30/3/2 (Q30a) & 30/3/3 (Q30a)
34. Prove the Identity Using Reciprocal Ratios
3 Marks
Prove that:
\[
\left(\frac1{\cos A}-\cos A\right)
\left(\frac1{\sin A}-\sin A\right)
=
\frac1{\tan A+\cot A}
\]
Series 30/3/2 (Q30b) & 30/3/3 (Q30b)
35. Prove the Identity Using Cubic Terms
3 Marks
Prove that:
\[
\frac{\cos\theta-2\cos^3\theta}
{\sin\theta-2\sin^3\theta}
+\cot\theta
=0
\]
Series 30/6/1 (Q30a) & 30/6/2 (Q26a)
36. Given Sum of Sine and Cosine
3 Marks
Given that
\[
\sin\theta+\cos\theta=x,
\]
prove that:
\[
\sin^4\theta+\cos^4\theta
=
\frac{2-(x^2-1)^2}{2}
\]
Series 30/6/1 (Q30b) & 30/6/2 (Q26b)