Class 11 RD Sharma Solutions – Chapter 9 Trigonometric Ratios of Multiple and Submultiple Angles – Exercise 9.1 | Set 3

Question 30(i). If 0 ≤ x≤ π and x lies in the 2nd quadrant such that sinx = 1/4, Find the values of cos(x/2), sin(x/2), and tan(x/2).

Solution:

Given that,

 sinx = 1/4

As we know that, sinx = √(1 – cos²x)

So, 

⇒ (1/4)² = (1 – cos²x)

⇒ (1/16) – 1 = – cos²x

cosx = ± √15/4

It is given that x is in 2nd quadrant, so cosx is negative.

cosx = – √15/4

Now,

As we know that, cosx = 2 cos²(x/2) – 1

So, 

⇒ – √15/4 = 2cos²(x/2) – 1 

⇒ cos²(x/2) = – √15/8 + 1/2

cos(x/2) = ± (4-√15)/8

It is given that, x is in 2nd quadrant, so cos(x/2) is positive.

cos(x/2) = (4 – √15)/8

Again,

cosx = cos²(x/2) – sin²(x/2)

⇒ – √15/4 = {(4 – √15)/8}² – sin²(x/2)

⇒ sin²(x/2) = (4 + √15)/8

⇒ sin(x/2) = ± √{(4 + √15)/8} = √{(4 + √15)/8}

Now,

tan(x/2) = sin(x/2) / cos(x/2)

= 

= 

= 

= 

= 

= 4 + √15

Hence, the value of cos(x/2) = (4 – √15)/8, sin(x/2) = √{(4 + √15)/8}, and tan(x/2) = 4 + √15 .

Question 30(ii). If cosx = 4/5 and x is acute, find tan2x.

Solution:

Given that,  

cosx = 4/5

As we know that, sinx = √(1 – cos²x)

So, 

= √(1 – (4/5)²)

= √(1 – 16/25)

= √{(25 – 16)/25}

= √(9/25)

= 3/5

Since, tanx = sinx/cosx, so

= (3/5) / (4/5)

= 3/4

As we know that,

tan2x = 2tanx / (1 – tan²x)

= 2(3/4) / {1 – (3/4)²}

= 2(3/4) / (1 – 9/16)

= (3/2) / (7/16)

= 24/7

Hence, the value of tan2x is 24/7

Question 30(iii). If sinx = 4/5 and 0 < x < π/2, then find the value of sin4x.

Solution:

Given that, 

sinx = 4/5

As we know that, sinx = √(1 – cos²x)

So, 

⇒ (4/5)² = 1 – cos²x

⇒ 16/25 – 1 = -cos²x

⇒ 9/25 = cos²x

⇒ cosx = ±3/5

It is given that, x is ln the 1st quadrant 

So, cosx = 3/5

Now,

sin4x = 2 sin2x cos2x

= 2 (2 sinx cosx)(1 – 2sin²x)

= 2(2 × 4/5 × 3/5)(1 – 2(4/5)²)

= 2(24/25)(1-32/25)

= 2(24/25)((25-32)/25)

= 2(24/25)(-7/25)

= -336/625

Hence, the value of sin4x is (- 336/625)

Question 31. If tanx = b/a, then find the value of 

Solution:

We have to find the value of 

So, 

= 

It is given that tanx = b/a, so

= 

= 

= 

= 

= 

= 

Hence, the value of  is 

Question 32. If tanA = 1/7 and tanB = 1/3, show that cos2A = sin4B

Solution:

Given that, tanA = 1/7 and tanB = 1/3

Show: cos2A = sin4B

As we know that, tan2B = 2tanB / (1 – tan²B)

= (2 × 1/3)(1 – 1/9) = 3/4

So, cos2A = (1 – tan²A)/(1 + tan²A)  

= {1-(1/7)²}/{1+(1/7)²}                     

= 48/50                                             

= 24/25

And sin4B = 2tan2B / (1 + tan²2B)

= {2 × 3/4}{1 + (3/4)²}

= 24/25

Hence, cos2A = sin4B

Question 33. cos7° cos14° cos28° cos56° = sin68°/16cos83°

Solution:

Lets solve LHS 

= cos7° cos14° cos28° cos56°

On dividing and multiplying by 2sin7°, we get

=  × 2sin7° × cos7° × cos14° × cos28° × cos56°

=  × cos28° × cos56°

=  × cos56°

= 

= 

= 

LHS = RHS

Hence proved.

Question 34. Proved that, cos(2π/15)cos(4π/15)cos(8π/15)cos(16π/15) = 1/16

Solution:

Let’s solve LHS 

= cos(2π/15)cos(4π/15)cos(8π/15)cos(16π/15)

On dividing and multiplying by 2sin(2π/15), we get

= 

= 

= 

= 

= 

= 

= 1/16 

LHS = RHS

Hence proved.

Question 35. Proved that, cos(π/5)cos(2π/5)cos(4π/5)cos(8π/5) = -1/16

Solution:

Lets solve LHS 

= cos(π/5)cos(2π/5)cos(4π/5)cos(8π/5)

On dividing and multiplying by 2sin(2π/5), we get

=  × 2sin(π/5)cos(π/5)cos(2π/5)cos(4π/5)cos(8π/5)

= (sin(2π/5)cos(2π/5)cos(4π/5)cos(8π/5))

= [2sin(2π/5)cos(2π/5)cos(4π/5)cos(8π/5)]

= [sin(4π/5)cos(4π/5)cos(8π/5)]

= [2sin(4π/5)cos(4π/5)cos(8π/5)]

= [sin(8π/5)cos(8π/5)]  

=[2sin(8π/5)cos(8π/5)]

= 

=

= 

= -1/16 

LHS = RHS

Hence proved.

Question 36. Proved that, cos(π/65)cos(2π/65)cos(4π/65)cos(8π/65)cos(16π/65)cos(32π/65) = 1/64

Solution:

Lets solve LHS 

= cos(π/65)cos(2π/65)cos(4π/65)cos(8π/65)cos(16π/65)cos(32π/65)

Now on dividing and multiplying by 2sin(π/65), we get

=  × 2sin(π/65)cos(π/65)cos(2π/65)cos(4π/65)cos(8π/65)cos(16π/65)cos(32π/65)

=  × [cos(2π/65) × cos(4π/65) × cos(8π/65) × cos(16π/65) × cos(32π/65)]

=  × cos(4π/65) × cos(8π/65) × cos(16π/65) × cos(32π/65)

=  × cos(8π/65) × cos(16π/65) × cos(32π/65)

=  × cos(16π/65) × cos(32π/65)

=  × cos(32π/65)

= 

= 

= 

= 1/64 

LHS = RHS

Hence proved

Question 37. If 2tanα = 3tanβ, prove that tan(α – β) = sin2β / (5 – cos2β)

Solution:

Given that,

2tanα = 3tanβ

Prove: tan(α – β) = sin2β / (5 – cos2β)

Proof:

Lets solve LHS 

= 

= 

= 

= 

= 

= 

= 

= 

= 

= 

= 

= 

= 

= 

LHS = RHS

Hence proved.

Question 38(i). If sinα + sinβ = a and cosα + cosβ = b, prove that sin(α + β) = 2ab/(a² + b²)

Solution:

Given that, 

sinα + sinβ = a and cosα + cosβ = b

Prove: sin(α + β) = 2ab/(a² + b²)

Proof:

As we know that, 

So       ……(i)

Now, using the identity

   …..(ii)

Now on dividing  eq(i) and (ii), we get

tan(α + β)/2 = a/b

As we know that,

sin2x = 2tanx/(1 + tan²x)

= 

= 2ab/(a² + b²)

LHS = RHS

Hence proved

Question 38(ii). If sinα + sinβ = a and cosα + cosβ = b, prove that cos(α – β) = (a² + b² – 2)/2

Solution:

Given that, 

sinα + sinβ = a ……(i) 

cosα + cosβ = b …….(ii)

Now on squaring eq(i) and (ii) and then adding them, we get

sin²α + sin²β + 2sinαsinβ + cos²α + cos²β + 2cosαcosβ = a² + b²

⇒ 1 + 1 + 2(sinαsinβ + cosαcosβ) = a² + b²

⇒ 2(sinαsinβ + cosαcosβ) = a² + b² – 2   

⇒ 2cos(α – β) = a² + b² – 2  

⇒ cos(α – β) = (a² + b² – 2)/2 

Hence proved.

Question 39. If 2tan(α/2) = tan(β/2), prove that cosα = 

Solution:

Given that, 

2tan(α/2) = tan(β/2)

Prove: cosα = 

Proof:

Let us solve RHS 

= 

= 

= 

= 

= 

= 

= 

= cosα 

RHS = LHS

Hence proved.

Question 40. If cosx = , prove that tan(x/2) = ± tan(α/2)tan(β/2).

Solution:

Given that,

    …..(i)

⇒ 

Now, by componendo and dividendo, we get

⇒ 

⇒ 

⇒ 

⇒ 

⇒ tan²(x/2) = tan²(α/2)tan²(β/2)

⇒ tan(x/2) = ±tan(α/2)tan(β/2)

Hence Proved.

Question 41. If sec(x + α) + sec(x – α) = 2secx, prove that cosx = ± √2 cos(α/2).

Solution:

Given that, 

sec(x + α) + sec(x – α) = 2secx

So, 

⇒ 

⇒ 

⇒ 

⇒ cos²xcosα = cos²x(cos²α + sin²α) – sin²α

⇒ cos²x(1 – cosα) = sin²α

⇒ 

= 

⇒ cosx = ± √2 cos(α/2)

Hence Proved

Question 42. If cosα + cosβ = 1/3 and sinα + sinβ = 1/4, prove that cos(α – β)/2 = ±5/24.

Solution:

Given that, 

cosα + cosβ = 1/3  

sinα + sinβ = 1/4, we get

Prove:  cos(α – β)/2 = ±5/24

Proof:

(cos²α + cos²β + cosαcosβ) + (sin²α + sin²β + 2sinαsinβ) = 1/9 + 1/16

1 + 1 + 2(cosαcosβ + sinαsinβ) = 25/144

2 + 2cos(α – β) = -263/288  …..(i)

Now,

=   [From (i)]

= 25/576

= ± 5/24

Hence proved.

Question 43. If sinα = 4/5 and cosβ = 5/13, prove that cos{(α – β)/2} = 8/√65.

Solution:

Given that,

sinα = 4/5 and cosβ = 5/13

As we know that.

cosα = √(1 – sin²α)

So, 

= √{1 – (4/5)²}

= 3/5

Also, sinβ = √(1 – cos²β)

= √{1 – (5/13)²}

= 12/13

Now,

cos(α – β) = cosα cosβ + sinα sinβ

= (3/5)(5/13)(4/5)(12/13)

= 63/65

Thus,

cos{(α – β)/2} = 

= 

= 8/√65

Hence Proved.

Question 44. If acos2θ + bsin2θ = c has α and β as its roots prove that,

(i) tanα + tanβ = 2b/(a + c)

(ii) tanα tanβ = (c – a)/(c + a)

(iii) tan(α + β) = b/a

Solution:

As we know that

Now substitute these values in the given equation, we get

a(1 – tan²θ) + b(2tanθ) = c(1 + tan²θ)

(c + a)tan²θ + 2btanθ + c – a = 0 

(i) As α and β are roots

So, sum of the roots:

tanα + tanβ = 2b / (c + a)

(ii) As α and β are roots

So, product of roots:

tanα tanβ = (c – a) / (c + a)

(iii) tan(α + β)= 

= 

= b/a

Hence proved.

Question 45. If cosα + cosβ = 0 = sinα + sinβ, then prove that cos2α + cos2β = -2cos(α + β).

Solution:

Given that,

 cosα + cosβ = 0 = sinα + sinβ

Prove: cos2α + cos2β = -2cos(α + β)

Proof:

cosα + cosβ = 0

On squaring on both sides, we get

cos²α + cos²β + 2 cosα cosβ = 0     ….(i)

Similarly

sinα + sinβ = 0

On squaring on both sides, we get

sin²α + sin²β + 2 sinα sinβ = 0       …..(ii)

Now, subtract eq (ii) from (i), we get

⇒ (cos²α + cos²β + 2 cosα cosβ) – (sin²α + sin²β + 2 sinα sinβ) = 0

⇒ cos²α – sin²α + cos²β – sin²β + 2(cosα cosβ – sinα sinβ) = 0

⇒ cos2α + cos2β + 2cos(α + β) = 0

⇒ cos2α + cos2β = -2cos(α + β)

Hence proved.