# R S AGGARWAL AND V AGGARWAL Solutions Mathematics Class 10 Chapter 8 Trigonometric Identities

## Chapter 8 – Trigonometric Identities Exercise Ex. 8A

Question 1

Prove the following identities:

Solution 1

(i)

LHS = RHS

(ii)

LHS = RHS

Question 2

Prove the following identities:

(i)

(ii)

(iii)

Solution 2

(i)

LHS = RHS

(ii)

LHS = RHS

(iii)

LHS = RHS

Question 3

Prove the following identities:

(i)

(ii)

Solution 3

(i)

LHS = RHS

(ii)

LHS = RHS

Question 4

Prove the following identities:

(i)

(ii)

Solution 4

(i)

(ii)

LHS = RHS

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Prove the following identities:

(i)

(ii)

Solution 9

(i)

LHS = RHS

(ii)

LHS = RHS

Question 10

Prove the following identities:

Solution 10

(i)    LHS =

(ii)

Hence, LHS = RHS

Question 11

Prove the following identity:

Solution 11

LHS = RHS

Question 12

Prove the following identity:

Solution 12

Question 13

Prove the following identity:

Solution 13

Question 14

Prove the following identity:

Solution 14

RHS = LHS

Question 15

Prove the following identity:

Solution 15

LHS =

RHS = LHS

Question 16

Prove the following identity:

Solution 16

LHS = RHS

Question 17

Prove the following identity:

Solution 17

RHS = LHS

Question 18

Prove the following identity:

Solution 18

Question 19

Prove the following identities:

Solution 19

(i)To prove

We know,

Therefore, LHS = RHS

(ii)

Therefore, LHS = RHS

(iii)

Question 20

Prove the following identity:

(i)

(ii)

Solution 20

(i)

LHS = RHS

(ii)

LHS = RHS

Question 21

Prove the following identities:

(i)

(ii)

Solution 21

(i)

LHS = RHS

(ii) LHS =

Question 22

Prove the following identities:

Solution 22

(i)

LHS =

Hence, LHS = RHS

(ii)

LHS = RHS

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Prove the following identity:

Solution 26

LHS = RHS

Question 27

Prove the following identity:

Solution 27

LHS = RHS

Question 28

Prove the following identities:

(i)

(ii)

Solution 28

(i)

LHS = RHS

(ii)

LHS = RHS

Question 29

Prove the following identity:

Solution 29

LHS = RHS

Question 30

Prove the following identities:

Solution 30

(i)

Further,

LHS = RHS

(ii)

LHS =

Further,

Question 31

Prove the following identities:

(i)

(ii)

Solution 31

(i)

On dividing the numerator and denominator of LHS by cos,We get

(ii)

On dividing the numerator and denominator of LHS by cos,We get

LHS = RHS

Question 32

Prove the following identity:

Solution 32

LHS = RHS

Question 33

Prove the following identity:

Solution 33

Question 34

Prove the following identity:

Solution 34

Question 35

Prove the following identity:

Solution 35

Question 36

Solution 36

Question 37

Prove the following identity:

Solution 37

Question 38

Prove the following identity:

Solution 38

Question 39

Prove the following identity:

Solution 39

Question 40

Show
that none of the following is an identity:

cos2
θ
+ cos θ = 1

Solution 40

Question 41

sin2
θ
+ sin θ
= 2

Solution 41

Question 42

tan2
θ
+ sin θ
= cos2 θ

Solution 42

Question 43

Prove
that:

(sin
θ
– 2sin3 θ)
= (2cos3 θ
cos θ)tan θ

Solution 43

## Chapter 8 – Trigonometric Identities Exercise Ex. 8B

Question 1

If a cos + b sin = m and a sin – b cos = n, prove that

.

Solution 1

Question 2

If x = a sec + b tan and y = a tan + b sec , prove that

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

If prove that

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

If
(cosec θ
– sin θ)
= a3 and (sec θ – cos θ) = b3, prove that a2b2
(a2 + b2) = 1.

Solution 9

Question 10

If
(2sin θ
+ 3cos θ)
= 2, prove that (3sin θ – 2cos θ) = ±
3.

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

If
sec θ
+ tan θ
= p, prove that

Solution 13

Question 14

If
sec θ
+ tan θ
= p, prove that

Solution 14

Question 15

If
sec θ
+ tan θ
= p, prove that

Solution 15

Question 16

Solution 16

Question 17

Solution 17

## Chapter 8 – Trigonometric Identities Exercise Ex. 8C

Question 1

Write
the value of (1 – sin2 θ) sec2 θ.

Solution 1

Question 2

Write
the value of (1 – cos2θ) cosec2 θ.

Solution 2

Question 3

Write
the value of (1 + tan2 θ) cos2 θ.

Solution 3

Question 4

Write
the value of (1 + cot2 θ) sin2 θ.

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Write
the value of sin θ cos (90– θ)
+ cos θ sin (90
– θ).

Solution 7

Question 8

Write
the value of cosec2 (90 – θ)
– tan2 θ.

Solution 8

Question 9

Write
the value of sec2 θ (1 + sin θ)(1 – sin θ).

Solution 9

Question 10

Write
the value of cosec2 θ(1 + cos θ)(1 – cos θ)

Note:
Question modified

Solution 10

Question 11

Write
the value of sin2 θ cos2 θ (1
+ tan2 θ)(1 + cot2 θ).

Solution 11

Question 12

Write
the value of (1 + tan2 θ)(1
+ sin θ)(1
– sin θ).

Solution 12

Question 13

Write
the value of 3cot2 θ – 3cosec2θ.

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Write
the value of tan 10ᵒ tan 20ᵒ
tan 70ᵒ
tan 80ᵒ.

Solution 27

Question 28

Write
the value of tan 1ᵒ tan 2ᵒ
… tan 89ᵒ.

Solution 28

Question 29

Write
the value of cos 1ᵒ
cos 2ᵒ…cos 180ᵒ.

Solution 29

Question 30

Solution 30

Question 31

If sin θ = cos (θ – 45ᵒ),  where θ is a acute, find the value of θ.

Solution 31

Question 32

Solution 32

Question 33

Find
the value of sin 48ᵒ sec 42ᵒ
+ cos 48ᵒ cosec 42
.

Solution 33

Question 34

If
x = a sin θ and y = b cos
θ,
write the value of (b2x2 + a2y2).

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

If sec θ + tan θ
= x , find the value of sec θ.

Solution 37

Question 38

Solution 38

Question 39

If
sin θ
= x, write the value of cot θ.

Solution 39

Question 40

If
sec θ
= x, write the value of tan θ.

Solution 40

## Chapter 8 – Trigonometric Identities Exercise FA

Question 1

Solution 1

Question 2

Solution 2

Question 3

If cos A + cos2
A = 1 then (sin2 A + sin4 A) =
?

(a)

(b) 2

(c) 1

(d) 4

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Prove that (sin 32° cos 58° + cos 32° sin 58°) = 1.

Solution 14

Question 15

If x = a sin θ + b cos
θ and y = a cos θ – b sin θ, prove that x2
+ y2 = a2 + b2.

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

If sec 5A = cosec (A – 36°) and 5A is an acute angle, show that A = 21°

Solution 20

## Chapter 8 – Trigonometric Identities Exercise MCQ

Question 1

Solution 1

Question 2

(a)0

(b) 1

(c) 2

(d) none of these

Solution 2

Question 3

tan 10° tan 15° tan 75° tan 80° = ?

Solution 3

Question 4

tan 5° tan 25° tan 30° tan 65° tan 85° = ?

Solution 4

Question 5

cos 1° cos 2° cos 3° …… cos 180° = ?

(a) -1

(b) 1

(c) 0

(d)

Solution 5

Question 6

Solution 6

Question 7

sin 47° cos 43° + cos 47° sin 43° = ?

(a) sin 4°

(b) cos 4°

(c) 1

(d) 0

Solution 7

Question 8

sec 70° sin 20° + cos 20° cosec 70° = ?

(a) 0

(b) 1

(c) -1

(d) 2

Solution 8

Question 9

If sin 3A = cos (A – 10o) and 3A is acute then A = ?

(a) 35°

(b) 25°

(c) 20°

(d) 45°

Solution 9

Question 10

If sec 4A = cosec (A – 10°) and 4A is acute then A = ?

(a) 20°

(b) 30°

(c) 40°

(d) 50°

Solution 10

Question 11

If A and B are acute angles such that sin A = cos B then (A + B) =?

(a) 45°

(b) 60°

(c) 90°

(d) 180°

Solution 11

Question 12

If cos (𝛼 + 𝛽) = 0 then sin (𝛼 – 𝛽) = ?

(a) sin 𝛼

(b) cos 𝛽

(c) sin 2𝛼

(d) cos 2𝛽

Solution 12

Question 13

sin (45° + θ) – cos (45°θ) = ?

(a) 2 sin θ

(b) 2 cos θ

(c) 0

(d) 1

Solution 13

Question 14

sec210° – cot2 80° = ?

(a) 1

(b) 0

Solution 14

Question 15

cosec2 57° – tan2 33° = ?

(a) 0

(b) 1

(c) -1

(d) 2

Solution 15

Question 16

Solution 16

Question 17

(a) 0

(b) 1

(c) 2

(d) 3

Solution 17

Question 18

(a) 0

(b) 1

(c) -1

(d) none of these

Solution 18

Question 19

Solution 19

Question 20

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Solution 20

Question 21

If 2cos 3θ = 1 then θ = ?

(a) 10°

(b) 15°

(c) 20°

(d) 30°

Solution 21

Question 22

(a) 15°

(b) 30°

(c) 45°

(d) 60°

Solution 22

Question 23

If tan x = 3cot x then x = ?

(a) 45°

(b) 60°

(c) 30°

(d) 15°

Solution 23

Question 24

If x tan 45° cos 60° = sin 60° cot 60° then x = ?

Solution 24

Question 25

If tan2 45° – cos2 30° = x sin 45° cos 45° then x = ?

Solution 25

Question 26

sec2 60° – 1 = ?

(a) 2

(b) 3

(c) 4

(d) 0

Solution 26

Correct option: (b)

sec2 60° – 1 = (2)2 – 1 = 4 – 1 = 3

Question 27

(cos 0° + sin 30° + sin 45°)(sin 90° + cos 60°cos 45°) =?

Solution 27

Question 28

sin230° + 4cot2 45° – sec2 60° = ?

Solution 28

Question 29

3cos2 60° + 2cot2 30° – 5sin2 45° = ?

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

If (tan θ + cot θ) = 5 then (tan2 θ + cot2 θ) = ?

(a) 27

(b) 25

(c) 24

(d) 23

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

If sin A + sin2 A = 1 then cos2 A + cos4 A = ?

(a)

(b) 1

(c) 2

(d) 3

Solution 47

Question 48

If cos A + cos2 A = 1 then sin2 A + sin4 A = ?

(a) 1

(b) 2

(c) 4

(d) 3

Solution 48

Question 49

(a) sec A + tan A

(b) sec A – tan A

(c) sec A tan A

(d) none of these

Solution 49

Question 50

(a) cosec A – cot A

(b) cosec A + cot A

(c) cosec A cot A

(d) none of these

Solution 50

Question 51

Solution 51

Question 52

(cosec θ – cot θ)2 = ?

Solution 52

Question 53

(sec A + tan A)(1 – sin A) = ?

(a) sin A

(b) cos A

(c) sec A

(d) cosec A

Solution 53

error: Content is protected !!