# Class 10 RD Sharma Solutions – Chapter 6 Trigonometric Identities – Exercise 6.2

**Question 1. If cos **θ** = 4/5, find all other trigonometric ratios of angle **θ**. **

**Solution:**

We are given, cos θ = 4/5. So, sec θ =1/cos θ = 5/4.

Now we know,

=> sin θ = √(1 – cos

^{2}θ)=> sin θ = √(1 – (4/5)

^{2})=> sin θ = √(1 – (16/25))

=> sin θ = √(9/25)

=> sin θ = 3/5

So, cosec θ = 1/sin θ = 5/3

And tan θ = sin θ/cos θ = (3/5)/(4/5) = 3/4

Therefore, cot θ = 1/tan θ = 4/3

If cos θ = 4/5, value of sec θ, sin θ, cosec θ, tan θ and cot θ are 5/4, 3/5, 5/3, 3/4 and 4/3 respectively.

**Question 2. If sin **θ** = 1/√2, find all other trigonometric ratios of angle **θ.

**Solution:**

We are given, sin θ = 1/√2. So, cosec θ =1/sin θ = √2.

Now we know,

=> cos θ = √(1 – sin

^{2}θ)=> cos θ = √(1 – (1/√2)

^{2})=> cos θ = √(1 – (1/2))

=> cos θ = √(1/2)

=> cos θ = 1/√2

So, sec θ = 1/cos θ = √2

And tan θ = sin θ/cos θ = (1/√2)/(1/√2) = 1

Therefore, cot θ = 1/tan θ = 1

If sin θ = 1/√2, value of cosec θ, cos θ, sec θ, tan θ and cot θ are √2, 1/√2, √2, 1 and 1 respectively.

**Question 3. If tan **θ** = 1/√2, find the value of ****.**

**Solution:**

We are given, tan θ = 1/√2. Now we know,

=> sec θ = √(1 + tan

^{2}θ)=> sec θ = √(1 + (1/√2)

^{2})=> sec θ = √(1+(1/2))

=> sec θ = √(3/2)

And cot θ = 1/tan θ = √2. Also, we know,

=> cosec θ = √(1 + cot

^{2}θ)=> cosec θ = √(1 + (√2)

^{2})=> cosec θ = √(1 + 2)

=> cosec θ = √3

So, =

=

=

Therefore, the value ofis.

**Question 4. If tan **θ** = 3/4, find the value of ****.**

**Solution:**

We are given, tan θ = 3/4. Now we know,

=> sec θ = √(1 + tan

^{2}θ)=> sec θ = √(1 + (3/4)

^{2})=> sec θ = √(1+(9/16))

=> sec θ = √(25/16)

=> sec θ = 5/4

And cos θ = 1/sec θ = 4/5.

So, =

=

=

Therefore, the value ofis.

**Question 5. If tan **θ** = 12/5, find the value of** **.**

**Solution:**

We are given, tan θ = 12/5. So, cot θ = 1/tan θ = 5/12.

Now we know,

=> cosec θ = √(1 + cot

^{2}θ)=> cosec θ = √(1 + (5/12)

^{2})=> cosec θ = √(1 + (25/144))

=> cosec θ = √(169/144)

=> cosec θ = 13/12

And sin θ = 1/cosec θ = 12/13.

So, =

=

= 25

Therefore, the value ofis 25.

**Question 6. If cot **θ** = 1/√3, find the value of ****.**

**Solution:**

We are given, cot θ = 1/√3. Now we know,

=> cosec θ = √(1 + cot

^{2}θ)=> cosec θ = √(1 + (1/√3)

^{2})=> cosec θ = √(1 + (1/3))

=> cosec θ = √(4/3)

=> cosec θ = 2/√3

And sin θ = 1/cosec θ = √3/2. Also, we know,

=> cos θ = √(1 – sin

^{2}θ)=> cos θ = √(1 – (√3/2)

^{2})=> cos θ = √(1 – (3/4))

=> cos θ = √(1/4)

=> cos θ = 1/2

So, =

=

=

=

Therefore, the value ofis.

**Question 7. If cosec A = √2, find the value of ****.**

**Solution:**

We are given, cosec A = √2. So, sin A = 1/cosec A = 1/√2.

Now we know,

=> cos A = √(1 – sin

^{2}A)=> cos A = √(1 – (1/√2)

^{2})=> cos A = √(1 – (1/2))

=> cos A = √(1/2)

=> cos A = 1/√2

Hence, tan A = sin A/cos A = (1/√2)/(1/√2) = 1. And cot A = 1/tan A = 1.

So, =

=

= 2

Therefore, the value ofis 2.

**Question 8. If cot **θ** = √3, find the value of ****.**

**Solution:**

We are given cot θ = √3. And tan θ = 1/cot θ = 1/√3.

Now we know,

=> cosec θ = √(1 + cot

^{2}θ)=> cosec θ = √(1 + (√3)

^{2})=> cosec θ = √4

=> cosec θ = 2

Also, we know,

=> sec θ = √(1 + tan

^{2}θ)=> sec θ = √(1 + (1/√3)

^{2})=> sec θ = √(1+(1/3))

=> sec θ = √(4/3)

=> sec θ = 2/√3

So, =

=

=

Therefore, the value ofis.

**Question 9. If 3cos **θ** = 1, find the value of** **.**

**Solution:**

We are given cos θ = 1/3. Now we know,

=> sin θ = √(1 – cos

^{2}θ)=> sin θ = √(1 – (1/3)

^{2})=> sin θ = √(1 – (1/9))

=> sin θ = √(8/9)

=> sin θ = 2√2/3

Hence, tan θ = sin θ/cos θ = (2√2/3)/(1/3) = 2√2

So, =

=

=

= 10

Therefore, the value ofis 10.

**Question 10. If √3 tan **θ = sin θ, find the value of sin^{2} θ – cos^{2} θ.** **

**Solution:**

We are given, √3 tan θ = sin θ

=> √3 (sin θ/cos θ) = sin θ

=> cos θ = 1/√3

Now we know,

=> sin θ = √(1 – cos

^{2}θ)=> sin θ = √(1 – (1/√3)

^{2})=> sin θ = √(1 – (1/3))

=> sin θ = √(2/3)

So, sin

^{2}θ – cos^{2}θ = √(2/3)^{2}– (1/√3)^{2}= 2/3 – 1/3

= 1/3

Therefore, the value of sin^{2}θ – cos^{2}θ is 1/3.

**Question 11. If cosec **θ** = 13/12, find the value of ****.**

**Solution:**

We are given, cosec θ = 13/12. So, sin θ = 1/cosec θ = 12/13.

Now we know,

=> cos θ = √(1 – sin

^{2}θ)=> cos θ = √(1 – (12/13)

^{2})=> cos θ = √(1 – (144/169))

=> cos θ = √(25/169)

=> cos θ = 5/13

So, =

=

=

= 3

Therefore, the value ofis 3.

**Question 12. If sin **θ + cos θ = √2 cos (90^{o}–θ), find cot θ.

**Solution:**

We are given,

=> sin θ + cos θ = √2 cos (90

^{o}–θ)=> sin θ + cos θ = √2 sin θ

=> cos θ = (√2–1)sin θ

=> cos θ/sin θ = √2–1

=> cot θ = √2–1

Therefore, value of cot θ is √2–1.

**Question 13. If 2sin**^{2} θ – cos^{2} θ = 2, then find the value of θ.

^{2}

**Solution:**

We have,

=> 2sin

^{2}θ – cos^{2 }θ = 2=> 2sin

^{2}θ – (1 – sin^{2}θ) = 2=> 2sin

^{2}θ – 1 + sin^{2}θ = 2=> 3sin

^{2}θ = 3=> sin

^{2}θ = 1=> sin θ = 1

=> θ = 90

^{o}

Therefore, the value of θ is 90^{o}.

**Question 14. If √3tan **θ – 1 = 0, find the value of sin^{2} θ – cos^{2} θ.

**Solution:**

We are given,

=> √3tan θ – 1 = 0

=> tan θ = 1/√3

=> tan θ = tan 30

^{o}=> θ = 30

^{o}So, sin

^{2}θ – cos^{2}θ = sin^{2}30^{o}– cos^{2}30^{o}= (1/2)

^{2}– (√3/2)^{2}= (1/4) – (3/4)

= –2/4

= –1/2

Therefore, the value of sin^{2}θ – cos^{2}θ is –1/2.

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