Introduction To Trigonometry

CBSE Class 10 — Maths — Previous Year Questions

Q1.VSA1 mark(2020)Easy

$\cos 80°+\cos 59°\cdot \csc 31°$ = ________.

Q2.VSA1 mark(2020)Easy

The value of $\left({\sin }^2\theta +\frac{1}{1+{\tan }^2\theta }\right)$ = ________.

Q3.VSA1 mark(2020)Easy

The value of $(1+{\tan }^2\theta )(1-\sin \theta )(1+\sin \theta )$ = ________.

Q4.VSA1 mark(2020)Easy

The value of $(\tan 1°\tan 2°\dots \tan 89°)$ is equal to ________.

Q5.VSA1 mark(2020)Easy

If $\sin A+{\sin }^{2}A=1$, then find the value of the expression $({\cos }^{2}A+{\cos }^{4}A)$.

Q6.VSA1 mark(2020)Easy

$\frac{3\cot 40°}{\tan 50°}-\frac{1}{2}\cdot \frac{\cos 35°}{\sin 55°}=$ ________.

Q7.VSA1 mark(2020)Easy

If $\cot \theta =\frac{7}{8}$, then the value of $\frac{(1+\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(1-\cos \theta )}$ is ________.

Q8.VSA1 mark(2020)Easy

What is the value of $\left(\frac{1}{1+{\cot }^2\theta }+\frac{1}{1+{\tan }^2\theta }\right)$?

Q9.MCQ1 mark(2020)Easy

The value of $\theta$ for which $\cos (10°+\theta )=\sin 30°$, is

Q10.VSA1 mark(2020)Easy

If $\tan A=1$, then $2\sin A\cos A=$ ________.

Q11.MCQ1 mark(2020)Easy

Simplest form of $\frac{1+{\tan }^{2}A}{1+{\cot }^{2}A}$ is ________.

Q12.VSA1 mark(2020)Easy

Evaluate: $\frac{2\tan 45°\times \cos 60°}{\sin 30°}$

Q13.MCQ1 mark(2023)Medium

If $2\tan A=3$, then the value of $\frac{4\sin A+3\cos A}{4\sin A-3\cos A}$ is
(A) $\frac{7}{\sqrt{13}}$
(B) $\frac{1}{\sqrt{13}}$
(C) $3$
(D) does not exist

Q14.MCQ1 mark(2023)Easy

$\left[\frac{3}{4}{\tan }^{2}30°-{\sec }^{2}45°+{\sin }^{2}60°\right]$ is equal to
(A) $-1$
(B) $\frac{5}{6}$
(C) $\frac{-3}{6}$
(D) $\frac{1}{6}$

Q15.MCQ1 mark(2023)Medium

$\sec \theta$ when expressed in terms of $\cot \theta$, is equal to:

Q16.MCQ1 mark(2023)Easy

If $\tan \theta =\frac{5}{12}$, then the value of $\frac{\sin \theta +\cos \theta }{\sin \theta -\cos \theta }$ is:

Q17.MCQ1 mark(2023)Easy

$\frac{2\tan 30°}{1+{\tan }^{2}30°}$ is equal to:

Q18.MCQ1 mark(2023)Easy

$\frac{{\cos }^2\theta }{{\sin }^2\theta }-\frac{1}{{\sin }^2\theta }$, in simplified form, is:

Q19.MCQ1 mark(2023)Easy

If $\theta$ is an acute angle of a right angled triangle, then which of the following equation is **not** true?

Q20.Assertion-Reason1 mark(2023)Medium

**Statement A (Assertion):** For $0<\theta \le 90°$, $\mathrm{cosec}\theta -\cot \theta$ and $\mathrm{cosec}\theta +\cot \theta$ are reciprocal of each other.

**Statement R (Reason):** ${\mathrm{cosec}}^2\theta -{\cot }^2\theta =1$

Q21.MCQ1 mark(2024)Medium

If $\sec \theta -\tan \theta =m$, then the value of $\sec \theta +\tan \theta$ is:

Q22.MCQ1 mark(2024)Easy

If $\cos (\alpha +\beta )=0$, then value of $\cos \left(\frac{\alpha +\beta }{2}\right)$ is equal to:

Q23.MCQ1 mark(2024)Easy

If $\sin \theta =\cos \theta$, $(0°<\theta <90°)$, then value of $(\sec \theta \cdot \sin \theta )$ is:

Q24.Assertion-Reason1 mark(2024)Easy

**Assertion (A):** If $\sin A=\frac{1}{3}$, $(0°<A<90°)$, then the value of $\cos A$ is $\frac{2\sqrt{2}}{3}$.

**Reason (R):** For every angle $\theta$, ${\sin }^2\theta +{\cos }^2\theta =1$.

Q25.MCQ1 mark(2024)Easy

If $\sin \theta =1$, then the value of $\frac{1}{2}\sin \frac{\theta }{2}$ is:

Q26.MCQ1 mark(2024)Easy

If $\frac{x}{3}=2\sin A$, $\frac{y}{3}=2\cos A$, then the value of $x^2+y^2$ is:

Q27.MCQ1 mark(2024)Easy

If $4\sec \theta -5=0$, then the value of $\cot \theta$ is:

Q28.MCQ1 mark(2025)Easy

If $\theta$ is an acute angle and $7+4\sin \theta =9$, then the value of $\theta$ is:

Q29.MCQ1 mark(2025)Easy

The value of ${\tan }^2\theta -\frac{1}{\cos \theta }\times \sec \theta$ is:

Q30.MCQ1 mark(2025)Easy

If $7{\cos }^2\theta +3{\sin }^2\theta =4$, then the value of $\theta$ is:

Q31.MCQ1 mark(2025)Easy

If $\sin 30°\tan 45°=\frac{\sec 60°}{k}$, then the value of $k$ is:

Q32.MCQ1 mark(2025)Easy

If $\tan 30=\sqrt{3}$, then $\frac{\theta }{2}$ equals:

Q33.MCQ1 mark(2025)Easy

$(\cot \theta +\tan \theta )$ equals:

Q34.MCQ1 mark(2025)Easy

If $\tan 2A=\sqrt{3}$ is true, then the measure of $\angle A$ is:

Q35.MCQ1 mark(2025)Easy

Which of the following statements is true?

Q36.VSA2 marks(2020)Medium

The rod $AC$ of a TV disc antenna is fixed at right angle to the wall $AB$ and a rod $CD$ is supporting the disc as shown in the given figure. If $AC=1.5$ m long and $CD=3$ m, find (i) $\tan \theta$ (ii) $\sec \theta +\csc \theta$.

Q37.VSA2 marks(2020)Medium

Prove that $1+\frac{{\cot }^2\alpha }{1+\csc \alpha }=\csc \alpha$.

Q38.VSA2 marks(2020)Medium

Show that ${\tan }^4\theta +{\tan }^2\theta ={\sec }^4\theta -{\sec }^2\theta$.

Q39.VSA2 marks(2020)Easy

If $\tan \theta =\frac{3}{4}$, find the value of $\left(\frac{1-{\cos }^2\theta }{1+{\cos }^2\theta }\right)$.

Q40.VSA2 marks(2020)Easy

If $\tan \theta =\sqrt{3}$, find the value of $\left(\frac{2\sec \theta }{1+{\tan }^2\theta }\right)$.

Q41.VSA2 marks(2020)Medium

Evaluate:
$${\left(\frac{\sin 47°}{\cos 43°}\right)}^2+{\left(\frac{\cos 30°}{\cot 30°}\right)}^2-(\sin 60°{)}^2$$

Q42.VSA2 marks(2020)Medium

If $A$, $B$ and $C$ are interior angles of a $△ABC$, then show that $\cos \left(\frac{B+C}{2}\right)=\sin \left(\frac{A}{2}\right)$.

Q43.VSA2 marks(2023)Medium

If $\sin \theta +\cos \theta =\sqrt{3}$, then find the value of $\sin \theta \cdot \cos \theta$.

Q44.VSA2 marks(2023)Medium

If $\sin \alpha =\frac{1}{\sqrt{2}}$ and $\cot \beta =\sqrt{3}$, then find the value of $\mathrm{cosec}\alpha +\mathrm{cosec}\beta$.

Q45.VSA2 marks(2023)Easy

If $4{\cot }^{2}45°-{\sec }^{2}60°+{\sin }^{2}60°+p=\frac{3}{4}$, then find the value of $p$.

Q46.VSA2 marks(2023)Medium

If $\cos A+{\cos }^{2}A=1$, then find the value of ${\sin }^{2}A+{\sin }^{4}A$.

Q47.VSA2 marks(2023)Medium

Evaluate: $\frac{5{\cos }^{2}60°+4{\sec }^{2}30°-{\tan }^{2}45°}{{\sin }^{2}30°+{\cos }^{2}30°}$

Q48.VSA2 marks(2023)Easy

If A and B are acute angles such that $\sin (A-B)=0$ and $2\cos (A+B)-1=0$, then find angles A and B.

Q49.VSA2 marks(2023)Medium

If $a\cos \theta +b\sin \theta =m$ and $a\sin \theta -b\cos \theta =n$, then prove that $a^2+b^2=m^2+n^2$.

Q50.VSA2 marks(2023)Medium

Prove that:
$$\sqrt{\frac{\sec A-1}{\sec A+1}}+\sqrt{\frac{\sec A+1}{\sec A-1}}=2\mathrm{cosec}A$$

Q51.VSA2 marks(2024)Easy

Evaluate: $2\sqrt{2}\cos 45°\sin 30°+2\sqrt{3}\cos 30°$

Q52.VSA2 marks(2024)Easy

If $A=60°$ and $B=30°$, verify that:
$\sin (A+B)=\sin A\cos B+\cos A\sin B$

Q53.VSA2 marks(2024)Easy

Evaluate: $2{\sin }^{2}30°\sec 60°+{\tan }^{2}60°$

Q54.VSA2 marks(2024)Easy

If $2\sin (A+B)=\sqrt{3}$ and $\cos (A-B)=1$, then find the measures of angles $A$ and $B$. $0\le A,B$, $(A+B)\le 90°$.

Q55.VSA2 marks(2024)Medium

Evaluate: $\frac{5{\cos }^{2}60°+4{\sec }^{2}30°-{\tan }^{2}45°}{{\sin }^{2}30°+{\sin }^{2}60°}$

Q56.VSA2 marks(2024)Easy

If $\sin (A-B)=\frac{1}{2}$, $\cos (A+B)=\frac{1}{2}$; $0<A+B\le 90°$, $A>B$; find $\angle A$ and $\angle B$.

Q57.VSA2 marks(2024)Easy

Evaluate:
$$\frac{2\tan 30°\cdot \sec 60°\cdot \tan 45°}{1-{\sin }^{2}60°}$$

Q58.VSA2 marks(2025)Easy

If $x\cos 60°+y\cos 0°+\sin 30°-\cot 45°=5$, then find the value of $x+2y$.

Q59.VSA2 marks(2025)Easy

Evaluate: $\frac{{\tan }^{3}60°}{{\sin }^{2}60°+{\cos }^{2}30°}$

Q60.VSA2 marks(2025)Easy

If $\tan A=\sqrt{3}$, where $A$ is an acute angle, then find the value of $\frac{1}{1+\cos A}$.

Q61.VSA2 marks(2025)Easy

If $4k={\tan }^{2}60°-2{\csc }^{2}30°-2{\tan }^{2}30°$, then find the value of $k$.

Q62.VSA2 marks(2025)Medium

Find the value of $x$ for which $(\sin A+\mathrm{cosec}A{)}^2+(\cos A+\sec A{)}^2=x+{\tan }^{2}A+{\cot }^{2}A$.

Q63.VSA2 marks(2025)Medium

Evaluate the following:
$$\frac{3\sin 30°-4{\sin }^{3}30°}{2{\sin }^{2}50°+2{\cos }^{2}50°}$$

Q64.SA3 marks(2020)Medium

If $\sin \theta +\cos \theta =\sqrt{3}$, then prove that $\tan \theta +\cot \theta =1$.

Q65.SA3 marks(2020)Medium

If $\sin \theta +\cos \theta =\sqrt{2}$, prove that $\tan \theta +\cot \theta =2$.

Q66.SA3 marks(2020)Medium

If $\sin \theta +\cos \theta =p$ and $\sec \theta +\csc \theta =q$, show that $q(p^2-1)=2p$.

Q67.SA3 marks(2020)Hard

Prove that:
$$\frac{\sin \theta -\cos \theta +1}{\cos \theta +\sin \theta -1}=\frac{1}{\sec \theta -\tan \theta }$$

Q68.SA3 marks(2020)Medium

Prove that: $\sqrt{\frac{1+\sin A}{1-\sin A}}=\sec A+\tan A$

Q69.SA3 marks(2023)Hard

Prove that: $\frac{\tan \theta +\sec \theta -1}{\tan \theta -\sec \theta +1}=\frac{1+\sin \theta }{\cos \theta }$

Q70.SA3 marks(2023)Medium

Prove that: $\left(\frac{1}{\cos \theta }-\cos \theta \right)\left(\frac{1}{\sin \theta }-\sin \theta \right)=\frac{1}{\tan \theta +\cot \theta }$

Q71.SA3 marks(2023)Medium

Prove that $\frac{\sin A-2{\sin }^{3}A}{2{\cos }^{3}A-\cos A}=\tan A$

Q72.SA3 marks(2023)Medium

Prove that $\sec A(1-\sin A)(\sec A+\tan A)=1$.

Q73.SA3 marks(2023)Hard

Prove that:
$$\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \mathrm{cosec}\theta$$

Q74.SA3 marks(2023)Medium

Prove that $\frac{1+\sec A}{\sec A}=\frac{{\sin }^{2}A}{1-\cos A}$.

Q75.SA3 marks(2024)Hard

Prove that: $\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \csc \theta$

Q76.SA3 marks(2024)Medium

Prove that $\frac{1+\sec \theta -\tan \theta }{1+\sec \theta +\tan \theta }=\frac{1-\sin \theta }{\cos \theta }$

Q77.SA3 marks(2024)Hard

Prove that $\frac{\sin \theta -\cos \theta +1}{\sin \theta +\cos \theta -1}=\frac{1}{\sec \theta -\tan \theta }$

Q78.SA3 marks(2024)Hard

Prove that:
$$\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}=1+\sec A\csc A$$

Q79.SA3 marks(2025)Hard

Prove that: $\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \cdot \csc \theta$

Q80.SA3 marks(2025)Hard

Prove that: $\frac{\sin A+\cos A}{\sin A-\cos A}+\frac{\sin A-\cos A}{\sin A+\cos A}=\frac{2}{2{\sin }^{2}A-1}$

Q81.SA3 marks(2025)Medium

Prove that: $\left(1+\frac{1}{{\tan }^2\theta }\right)\left(1+\frac{1}{{\cot }^2\theta }\right)=\frac{1}{{\sin }^2\theta -{\sin }^4\theta }$

Q82.SA3 marks(2025)Medium

Prove that: $\sqrt{\frac{\csc \theta -1}{\csc \theta +1}}+\sqrt{\frac{\csc \theta +1}{\csc \theta -1}}=2\sec \theta$

Q83.SA3 marks(2025)Medium

Prove that: $\sqrt{\frac{\sec A-1}{\sec A+1}}+\sqrt{\frac{\sec A+1}{\sec A-1}}=2\csc A$

Q84.SA3 marks(2025)Medium

Prove that: $\left(\frac{1}{\cos A}-\cos A\right)\left(\frac{1}{\sin A}-\sin A\right)=\frac{1}{\tan A+\cot A}$

Q85.SA3 marks(2025)Hard

Prove that $\frac{\cos A+\sin A-1}{\cos A-\sin A+1}=\mathrm{cosec}A-\cot A$.

Q86.SA3 marks(2025)Hard

If $\cot \theta +\cos \theta =p$ and $\cot \theta -\cos \theta =q$, prove that $p^2-q^2=4\sqrt{pq}$.