Class 10 Maths · Chapter 8

Introduction to Trigonometry

454 Class 10 Maths questions on Introduction to Trigonometry from the NCERT textbook, NCERT Exemplar and CBSE board papers 2018–2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 18 of 23

1. Q341 CBSE 2023
   Prove ​that $\frac{\sin A-2{\sin }^{3}A}{2{\cos }^{3}A-\cos A}=\tan A$.
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   Identity proved: LHS $=\tan A$.

   Step-by-step solution →
2. Q342 CBSE 2023
   Prove ​that : $\left(1+\frac{1}{{\tan }^{2}A}\right)\left(1+\frac{1}{{\cot }^{2}A}\right)=\frac{1}{{\sin }^{2}A-{\sin }^{4}A}$
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   Both sides reduce to $\frac{1}{{\sin }^{2}A-{\sin }^{4}A}$. Hence proved.

   Step-by-step solution →
3. Q343 CBSE 2023
   Prove ​that $\sec A(1-\sin A)(\sec A+\tan A)=1$.
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   Identity proved: LHS $=1$.

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4. Q344 CBSE 2023
   Prove that ​: $(\mathrm{cosec}A-\sin A)(\sec A-\cos A)(\tan A+\cot A)=1$.
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   LHS $=1=$ RHS. Hence proved.

   Step-by-step solution →
5. Q345 CBSE 2023
   Prove ​that ​: $(\mathrm{cosec}A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
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   Both sides reduce to $\sin A\cos A$, so the identity holds.

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6. Q346 CBSE 2023
   Prove that ​: $\frac{\tan \theta +\sec \theta -1}{\tan \theta -\sec \theta +1}=\frac{1+\sin \theta }{\cos \theta }$.
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   LHS simplifies to $\frac{1+\sin \theta }{\cos \theta }=$ RHS, proving the identity.

   Step-by-step solution →
7. Q347 CBSE 2023
   Prove ​that $\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}=1+\sec A\csc A$.
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   Identity proved: LHS $=1+\sec A\csc A$.

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8. Q348 CBSE 2023
   Prove ​that $\frac{1+\sec A}{\sec A}=\frac{{\sin }^{2}A}{1-\cos A}$.
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   Both sides equal $1+\cos A$, proving the identity.

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9. Q349 CBSE 2023
   Prove ​that ​: $\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \mathrm{cosec}\theta$.
   Show answer

   $1+\sec \theta \mathrm{cosec}\theta$ (proved)

   Step-by-step solution →
10. Q350 CBSE 2023
    Prove ​that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{{\cos }^{2}A}{(1+\sin A{)}^2}$.
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    Identity proved: LHS $=$ RHS.

    Step-by-step solution →
11. Q351 CBSE 2023
    Prove ​that $(\sec \theta +\tan \theta )(1-\sin \theta )=\cos \theta$.
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    Identity proved: LHS $=\cos \theta =$ RHS.

    Step-by-step solution →
12. Q352 CBSE 2023
    1. (i)Prove that ​: $\sqrt{{\sec }^2\theta +{\csc }^2\theta }=\tan \theta +\cot \theta$.
    2. (ii)​Evaluate : $\frac{\cos 45^{\circ }}{\sec 30^{\circ }+\csc 30^{\circ }}$.
    Show answer

    (i) Identity proved. (ii) $\frac{\cos 45^{\circ }}{\sec 30^{\circ }+\csc 30^{\circ }}=\frac{3-\sqrt{3}}{4\sqrt{2}}$.

    Step-by-step solution →
13. Q353 CBSE 2023
    If $x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ ​and $x\sin \theta =y\cos \theta$, prove ​that $x^2+y^2=1$.
    Show answer

    $x=\cos \theta$, $y=\sin \theta$, hence $x^2+y^2=1$.

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14. Q354 CBSE 2020
    If ​in $△ABC$, $\angle A=90^{\circ }$, then ​the ​value ​of $\mathrm{cosec}(B+C)$ ​is

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

    Step-by-step solution →
15. Q355 CBSE 2020
    The value ​of $\theta$ ​for ​which $\cos (10^{\circ }+\theta )=\sin 30^{\circ }$, ​is

    1. (a)$50^{\circ }$
    2. (b)$40^{\circ }$
    3. (c)$80^{\circ }$
    4. (d)$20^{\circ }$
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    (a) $50^{\circ }$

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16. Q356 CBSE 2020
    The ​value ​of $\sin \theta \cos (90^{\circ }-\theta )+\cos \theta \sin (90^{\circ }-\theta )$ ​is

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

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17. Q357 CBSE 2020
    If $\sin A=\cos A$, $0\le A\le 90^{\circ }$, ​then the ​angle $A$ is equal ​to

    1. (a)$30^{\circ }$
    2. (b)$60^{\circ }$
    3. (c)$0^{\circ }$
    4. (d)$45^{\circ }$
    Show answer

    (d) $45^{\circ }$

    Step-by-step solution →
18. Q358 CBSE 2020
    If $\sin \theta =\frac{3}{5}$, ​then ​the value ​of ​the expression $\frac{5\sin \theta -2\cos \theta }{\tan \theta }$ ​is

    1. (a)$\frac{15}{28}$
    2. (b)$\frac{28}{15}$
    3. (c)$\frac{23}{15}$
    4. (d)$\frac{92}{15}$
    Show answer

    (b) $\frac{28}{15}$

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19. Q359 CBSE 2020
    Simplest ​form of $\frac{1+{\tan }^{2}A}{1+{\cot }^{2}A}$ ​is ​________.
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    $\frac{1+{\tan }^{2}A}{1+{\cot }^{2}A}={\tan }^{2}A$

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20. Q360 CBSE 2020
    Given ​that in $△ABC$, $\angle B=90^{\circ }$, $\tan A=\frac{1}{\sqrt{3}}$, then ​the ​value ​of $\sin C$ ​is ​________.
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    $\sin C=\frac{\sqrt{3}}{2}$.

    Step-by-step solution →