Trigonometrical Identities - ICSE STD 10 Mathematics Questions

Q 1[3 marks sum]3 Marks

Simplify: $\frac{\sin A}{\cot A+\operatorname{cosec} A}-\frac{\sin A}{\cot A-\operatorname{cosec} A}$

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Q 2[3 marks sum]3 Marks

Find the value : $\tan ^2 \mathrm{~A}+\cot ^2 \mathrm{~A}-\sec ^2 \mathrm{~A} \operatorname{cosec}^2 \mathrm{~A}$

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Q 3[3 marks sum]3 Marks

For what value of $\alpha, 2 \sin ^2 \alpha-\cos ^2 \alpha=2$ ?

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Q 4[3 marks sum]3 Marks

Simplify : $\left(\sec ^2 \theta-2 \tan ^2 \theta\right)(1-\sin \theta)(1+\sin \theta)$

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Q 5[3 marks sum]3 Marks

Prove that: $\left(\sin ^2 \mathrm{~A} \cos ^2 \mathrm{~B}-\cos ^2 \mathrm{~A} \sin ^2 \mathrm{~B}\right)=\sin ^2 \mathrm{~A}-\sin ^2 \mathrm{~B}$.

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Q 6[4 marks sum]4 Marks

Prove that $(1+\cot \theta-\operatorname{cosec} \theta)(1+\tan \theta+\sec \theta)=2$.

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Q 7[4 marks sum]4 Marks

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

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Q 8[4 marks sum]4 Marks

If $x=a \sec \alpha \cos \beta, y=b \sec \alpha \sin \beta$ and $z=c \tan \alpha$, then evaluate $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}$.

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Q 9[4 marks sum]4 Marks

If $x=a \sec \theta+b \tan \theta$ and $y=a \tan \theta+b \sec \theta$, then show that $\frac{a^2-b^2}{x^2-y^2}$.

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Q 10[4 marks sum]4 Marks

Prove that : $\frac{(\sec \theta+\tan \theta-1)(\sec \theta-\tan \theta+1)}{\tan \theta}=2$

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Q 11MCQ1 Mark

Simplified form of $\frac{3-\tan \theta}{3 \operatorname{cosec} \theta-\sec \theta}$ is :

A
$\cos \theta$
B
$\sin \theta$
C
$\operatorname{cosec} \theta$
D
$\tan \theta$

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Q 12MCQ1 Mark

$2 \cos ^2 \theta+\frac{2}{1+\cot ^2 \theta}=$

A
1
B
2
C
0
D
$\frac{1}{2}$

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Q 13MCQ1 Mark

$(\sec \theta+\tan \theta)(1-\sin \theta)=$

A
$\sec \theta$
B
$\sin \theta$
C
$\operatorname{cosec} \theta$
D
$\cos \theta$

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Q 14MCQ1 Mark

$\frac{1+\tan ^2 \theta}{1+\cot ^2 \theta}=$

A
$\sec ^2 \theta$
B
-1
C
$\cot ^2 \theta$
D
$\tan ^2 \theta$

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Q 15MCQ1 Mark

$(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)=$

A
0
B
1
C
2
D
-1

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Q 16Assertion (A) & Reason (B) MCQ1 Mark

Assertion (A) : If $2 \tan \theta=4 \sin \theta$, then $\sin \theta=\cos \theta$.
Reason (R) : For any acute angle $\theta, \sin ^2 \theta \cdot \operatorname{cosec}^2 \theta=1$.

A
A is true, R is false.
✓
A is false, R is true.
C
Both A and R are true, and R is the correct reason for A .
D
Both A and R are true, and R is incorrect reason for A .

Answer: B.

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Q 17Assertion (A) & Reason (B) MCQ1 Mark

Assertion (A) : $\left(1-\operatorname{cosec}^2 \theta\right)\left(1-\sec ^2 \theta\right)=1$.
Reason (R) : $1+\tan ^2 \theta=\sec ^2 \theta$ and $1+\cot ^2 \theta=\operatorname{cosec}^2 \theta$.

A
A is true, R is false.
B
A is false, R is true.
✓
Both A and R are true, and R is the correct reason for A .
D
Both A and R are true, and R is incorrect reason for A .

Answer: C.

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Q 18Assertion (A) & Reason (B) MCQ1 Mark

Assertion (A) : $\sin \theta$ can be expressed in terms of $\sec \theta$ as : $\sin \theta=\frac{\sqrt{\sec ^2 \theta-1}}{\sec \theta}$
Reason (R) : For an acute angle $\theta, 1+\tan ^2 \theta=\sec ^2 \theta$.

A
A is true, R is false.
B
A is false, R is true.
✓
Both A and R are true, and R is the correct reason for A .
D
Both A and R are true, and R is incorrect reason for A .

Answer: C.

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Q 19Assertion (A) & Reason (B) MCQ1 Mark

Assertion (A) : If $x=2 \sin ^2 \mathrm{~A}, y=2 \cos ^2 \mathrm{~A}+1$, then the value of $x+y$ is 3 .
Reason (R) : For an acute angle, $\theta, \sin ^2 \theta+\cos ^2 \theta=1$.

A
A is true, R is false.
B
A is false, R is true.
✓
Both A and R are true, and R is the correct reason for A .
D
Both A and R are true, and R is incorrect reason for A .

Answer: C.

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Trigonometrical Identities question types

Trigonometrical Identities questions

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Trigonometrical Identities Mathematics - Trigonometrical Identities

Trigonometrical Identities question types

Trigonometrical Identities questions

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