Class 10 Maths · Chapter 8

Introduction to Trigonometry

108 Class 10 Maths questions on Introduction to Trigonometry from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 2 of 6

1. Q21 Exemplar · Exercise 8.1
   The ​value ​of $(\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }\dots \tan 89^{\circ })$ ​is

   1. (a)$0$
   2. (b)$1$
   3. (c)$2$
   4. (d)$\frac{1}{2}$
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   (b) $1$

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2. Q22 Exemplar · Exercise 8.1
   If $\cos 9\alpha =\sin \alpha$ ​and $9\alpha <90^{\circ }$, ​then ​the value of $\tan 5\alpha$ ​is

   1. (a)$\frac{1}{\sqrt{3}}$
   2. (b)$\sqrt{3}$
   3. (c)$1$
   4. (d)$0$
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   (c) $1$

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3. Q23 Exemplar · Exercise 8.1
   If $△ABC$ ​is ​right ​angled ​at $C$, ​then the value of $\cos (A+B)$ ​is

   1. (a)$0$
   2. (b)$1$
   3. (c)$\frac{1}{2}$
   4. (d)$\frac{\sqrt{3}}{2}$
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   (a) $0$

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4. Q24 Exemplar · Exercise 8.1
   If $\sin A+{\sin }^{2}A=1$, then ​the ​value of ​the ​expression $({\cos }^{2}A+{\cos }^{4}A)$ ​is

   1. (a)1
   2. (b)$\frac{1}{2}$
   3. (c)2
   4. (d)3
   Show answer

   (a) 1

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5. Q25 Exemplar · Exercise 8.1
   Given ​that $\sin \alpha =\frac{1}{2}$ and $\cos \beta =\frac{1}{2}$, ​then ​the value ​of $(\alpha +\beta )$ ​is

   1. (a)$0^{\circ }$
   2. (b)$30^{\circ }$
   3. (c)$60^{\circ }$
   4. (d)$90^{\circ }$
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   (d) $90^{\circ }$

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6. Q26 Exemplar · Exercise 8.1
   The ​value of ​the ​expression $\left[\frac{{\sin }^{2}22^{\circ }+{\sin }^{2}68^{\circ }}{{\cos }^{2}22^{\circ }+{\cos }^{2}68^{\circ }}+{\sin }^{2}63^{\circ }+\cos 63^{\circ }\sin 27^{\circ }\right]$ is

   1. (a)3
   2. (b)2
   3. (c)1
   4. (d)0
   Show answer

   (b) 2

   Step-by-step solution →
7. Q27 Exemplar · Exercise 8.1
   If $4\tan \theta =3$, ​then $\left(\frac{4\sin \theta -\cos \theta }{4\sin \theta +\cos \theta }\right)$ ​is equal to

   1. (a)$\frac{2}{3}$
   2. (b)$\frac{1}{3}$
   3. (c)$\frac{1}{2}$
   4. (d)$\frac{3}{4}$
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   (c) $\frac{1}{2}$

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8. Q28 Exemplar · Exercise 8.1
   If $\sin \theta -\cos \theta =0$, ​then the ​value of $({\sin }^4\theta +{\cos }^4\theta )$ ​is

   1. (a)$1$
   2. (b)$\frac{3}{4}$
   3. (c)$\frac{1}{2}$
   4. (d)$\frac{1}{4}$
   Show answer

   (c) $\frac{1}{2}$

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9. Q29 Exemplar · Exercise 8.1
   $\sin (45^{\circ }+\theta )-\cos (45^{\circ }-\theta )$ is ​equal ​to

   1. (a)$2\cos \theta$
   2. (b)$0$
   3. (c)$2\sin \theta$
   4. (d)$1$
   Show answer

   (b) $0$

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10. Q30 Exemplar · Exercise 8.2
    State ​whether ​the following ​is true ​or ​false: $\frac{\tan 47^{\circ }}{\cot 43^{\circ }}=1$.
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    True

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11. Q31 Exemplar · Exercise 8.2
    The ​value of the ​expression $({\cos }^{2}23^{\circ }-{\sin }^{2}67^{\circ })$ ​is ​positive.
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    False

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12. Q32 Exemplar · Exercise 8.2
    The ​value ​of ​the expression $(\sin 80^{\circ }-\cos 80^{\circ })$ is ​negative.
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    False

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13. Q33 Exemplar · Exercise 8.2
    $\sqrt{(1-{\cos }^2\theta ){\sec }^2\theta }=\tan \theta$
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    True

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14. Q34 Exemplar · Exercise 8.2
    If $\cos A+{\cos }^{2}A=1$, ​then ${\sin }^{2}A+{\sin }^{4}A=1$.
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    True

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15. Q35 Exemplar · Exercise 8.2
    State ​whether ​the ​following is true ​or ​false ​and ​justify: $(\tan \theta +2)(2\tan \theta +1)=5\tan \theta +{\sec }^2\theta$.
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    False

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16. Q36 Exemplar · Exercise 8.2
    State whether the following are true ​or ​false and justify: (9) The value of $2\sin \theta$ ​can be $a+\frac{1}{a}$, where $a$ is a positive number and $a\ne 1$. (10) $\cos \theta =\frac{a^2+b^2}{2ab}$, where $a$ and $b$ are ​two ​distinct ​numbers ​such ​that $ab>0$.

    1. (9)2 sin theta = a + 1/a, a > 0, a != 1
    2. (10)cos theta = (a^2 + b^2)/(2ab), a, b distinct, ab > 0
    Show answer

    Both statements are **false**. By AM-GM, $a+\frac{1}{a}>2\ge 2\sin \theta$ and $\frac{a^2+b^2}{2ab}>1\ge \cos \theta$.

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17. Q37 Exemplar · Exercise 8.2
    Write ​'True' or ​'False' and justify ​your answer: $\cos \theta =\frac{a^2+b^2}{2ab}$, ​where $a$ and $b$ are ​two ​distinct numbers such that $ab>0$.
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    False

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18. Q38 Exemplar · Exercise 8.3
    Show ​that ${\tan }^4\theta +{\tan }^2\theta ={\sec }^4\theta -{\sec }^2\theta$.
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    Proved: ${\sec }^4\theta -{\sec }^2\theta ={\sec }^2\theta ({\sec }^2\theta -1)=(1+{\tan }^2\theta ){\tan }^2\theta ={\tan }^4\theta +{\tan }^2\theta$.

    Step-by-step solution →
19. Q39 Exemplar · Exercise 8.3
    Prove ​that $\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }=2\mathrm{cosec}\theta$.
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    LHS simplifies to $\frac{2}{\sin \theta }=2\mathrm{cosec}\theta$, proving the identity.

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20. Q40 Exemplar · Exercise 8.3
    Prove ​that $\frac{\tan A}{1+\sec A}-\frac{\tan A}{1-\sec A}=2\mathrm{cosec}A$.
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    LHS simplifies to $2\mathrm{cosec}A$, which equals the RHS. Hence the identity is proved.

    Step-by-step solution →