Sample QuestionsTrigonometric Identities questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
$\frac{1+\tan^2\text{A}}{1+\cot^2\text{A}}$ is equal to:
- A
$\sec^2\text{A}$
- B
$-1$
- C
$\cot^2\text{A}$
- ✓
$\tan^2\text{A}$
Answer: D.
View full solution →$2(\sin^6\theta+\cos^6\theta)-3(\sin^4\theta+\cos^4\theta)$ is equal to:
Answer: C.
View full solution →If $\text{a}\cos\theta+\text{b}\sin\theta=4\text{ and a}\sin\theta-\text{b}\cos\theta=3,$ then $a^2+b^2=0$
Answer: C.
View full solution →If $\text{x}=\text{r}\sin\theta\cos\phi,\text{y}=\text{r}\sin\phi$ and ${z}=\text{r}\cos\theta,$ then:
- ✓
$x^2+y^2+z^2=r^2$
- B
$x^2+y^2-z^2=r^2$
- C
$x^2-y^2+z^2=r^2$
- D
$z^2+y^2-x^2=r^2$
Answer: A.
View full solution →$(\sec\text{A}+\tan\text{A})(1-\sin\text{A})=$
- A
$\sec\text{A}$
- B
$\sin\text{A}$
- C
$\text{cosec A}$
- ✓
$\cos\text{A}$
Answer: D.
View full solution →Statement-1 (A): For $0<\theta \leq 90^{\circ}, \operatorname{cosec} \theta-\cot \theta$ and $\operatorname{cosec} \theta+\cot \theta$ are reciprocal of each other.
Statement-2 (R): $\cot ^2 \theta-\operatorname{cosec}^2 \theta=1$
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
Answer: C.
View full solution →Statement-1 (A): If $\tan \theta+\cot \theta=2$, then $\tan ^2 \theta+\cot ^2 \theta=4$.
Statement-2 (R): If $\operatorname{cosec} A=\sqrt{2}$, then $\frac{2 \sin ^2 A+3 \cot ^2 A}{4 \tan ^2 A-2 \cos ^2 A}=\frac{4}{3}$.
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- ✓
Statement-1 is False, Statement-2 is True.
Answer: D.
View full solution →Statement-1 (A): If $\sin \theta+\sin ^2 0=1$, then $\cos ^2 0+\cos ^4 0=1$
Statement-2 (R): $1-\sin ^2 0=\cos ^2 0$.
- ✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
Answer: A.
View full solution →Statement-1 (A): Let $a, b$ be non-zero real numbers. Then, $\sec ^2 \theta=\frac{4 a b}{(a+b)^2}$ is true if and only if $a=b$.
Statement-2 (R): $\sec ^2 \theta \geq 1$ for $0 \leq \theta<90^{\circ}$.
- ✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- C
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
Answer: A.
View full solution →Statement-1 (A) : For $0 < \theta < 90^{\circ}, \sec \theta+\tan \theta$ and $\sec \theta-\tan \theta$ are reciprocal of each other.
Statement-2 (R): $\tan ^2 \theta-\sec ^2 \theta=1$
- A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- ✓
Statement-1 is True, Statement-2 is False.
- D
Statement-1 is False, Statement-2 is True.
Answer: C.
View full solution →Write 'True' or 'False' and justify your answer in the following:
The value of $\sin\theta+\cos\theta$ is always greater than 1.
View full solution →Write 'True' or 'False' and justify your answer in the following:
The value of the expression $\sin80^\circ-\cos80^\circ$ is negative.
View full solution →Write 'True' or 'False' and justify your answer in the following:
$\cos\theta=\frac{\text{a}^2+\text{b}^2}{2\text{ab}}$, where a and b are two lab distinct numbers such that ab > 0.
View full solution →Write 'True' or 'False' and justify your answer in the following:
The value of $\sin\theta$ is $\text{x}+\frac{1}{\text{x}},$ where 'x' is a positive real number.
View full solution →Write 'True' or 'False' and justify your answer in the following:
The value of $\cos^2{23}-\sin^2{67}$ is positive.
View full solution →If $\sin ^4 A-\cos ^4 A=1$ and $0 < A \leq 90^{\circ}$, then A = ____________ .
View full solution →If $2 \sin \theta+3 \cos \theta=2$, then $3 \sin \theta-2 \cos \theta$ = ____________ .
View full solution →If $(\tan \theta+2)(2 \tan \theta+1)=A \tan \theta+B \sec ^2 \theta$, then AB = ____________ .
View full solution →$\sqrt{-4+\sqrt{8+16 \operatorname{cosec}^4 \theta+\sin ^4 \theta}}=A \operatorname{cosec} \theta+B \sin \theta$, then A = ____________ and B = ____________ .
View full solution →The value of $(\operatorname{cosec} \theta-\sin \theta)(\sec \theta-\cos \theta)(\tan \theta+\cot \theta)$ is ____________ .
View full solution →What is the value of $\left(1+\tan ^2 \theta\right)(1-\sin \theta)(1+\sin \theta)$?
View full solution →What is the value of $\sin ^2 \theta+\frac{1}{1+\tan ^2 \theta}?$
View full solution →If $\operatorname{cosec} \theta=2 x$ and $\cot \theta=\frac{2}{x}$, find the value of $2\left(x^2-\frac{1}{x^2}\right)$.
View full solution →If $5 x=\sec \theta$ and $\frac{5}{x}=\tan \theta$, find the value of $5\left(x^2-\frac{1}{x^2}\right)$.
View full solution →If $\sin \theta=\frac{1}{3}$, then find the value of $2 \cot ^2 \theta+2$.
View full solution →Prove the following trigonometric identities.
$\text{cos}^2\text{A}+\frac{1}{1+\cot^2\text{A}}=1$
View full solution →Prove the following trigonometric identities.
$\sin^2\text{A}\cot^2\text{A}+\cos^2\text{A}\tan^2\text{A}=1$
View full solution →What is the value of $\frac{\tan^2\theta-\sec^2\theta}{\cot^2\theta-\text{cosec}^2\theta}?$
View full solution →Prove the following trigonometric identities.
$(1+\tan^2\theta)(1-\sin\theta)(1+\sin\theta)=1$
View full solution →Prove the following trigonometric identities.
$\frac{1+\cos\theta-\sin^2\theta}{\sin\theta(1+\cos\theta)}=\cot\theta$
View full solution →Prove the following trigonometric identities.
$\frac{1+\cos\theta+\sin\theta}{1+\cos\theta-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$
View full solution →If $\cos\text{A}=\frac{7}{25},$ find is the value of $\tan\text{A}+\cot\text{A}$.
View full solution →Prove the following trigonometric identities.
$\frac{\cot\text{A}+\tan\text{B}}{\cot\text{B}+\tan\text{A}}=\cot\text{A}\tan\text{B}$
View full solution →If $\cot\theta=\frac{1}{\sqrt{3}},$ find the value of $\frac{1+\cos^2\theta}{2-\sin^2\theta}$.
View full solution →If $\text{cosec A}=\sqrt{2},$ find the value of $\frac{2\sin^2\text{A}+3\cot^2\text{A}}{4(\tan^2-\cos^2\text{A})}$.
View full solution →Prove the following trigonometric identities.
$\frac{\sin\text{A}}{\sec\text{A}+\tan\text{A}-1}+\frac{\cos\text{A}}{\text{cosec A}+\cot\text{A}-1}=1$
View full solution →Prove the following trigonometric identities.
$\frac{\cot^2\text{A}(\sec\text{A}-1)}{1+\sin\text{A}}=\sec^2\text{A}\Big(\frac{1-\sin\text{A}}{1+\sin\text{A}}\Big)$
View full solution →Prove the following trigonometric identities.
$\frac{\tan^2\text{A}}{1+\tan^2\text{A}}+\frac{\cot^2\text{A}}{1+\cot^2\text{A}}=1$
View full solution →Prove the following trigonometric identities.
If $\text{T}_\text{n}=\sin^\text{n}\theta+\cos^\text{n}\theta,$ porve that $\frac{\text{T}_3-\text{T}_5}{\text{T}_1}=\frac{\text{T}_5-\text{T}_7}{\text{T}_3}.$
View full solution →Prove the following trigonometric identities.
$(\text{cosec }\theta-\sec\theta)(\cot\theta-\tan\theta)=(\text{cosec }\theta+\sec\theta)(\sec\theta\text{ cosec }\theta-2)$
View full solution →