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

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

**Solution:**

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

Now we know,

=> sin Î¸ = âˆš(1 â€“ cos

^{2}Î¸)=> sin Î¸ = âˆš(1 â€“ (4/5)

^{2})=> sin Î¸ = âˆš(1 â€“ (16/25))

=> sin Î¸ = âˆš(9/25)

=> sin Î¸ = 3/5

So, cosec Î¸ = 1/sin Î¸ = 5/3

And tan Î¸ = sin Î¸/cos Î¸ = (3/5)/(4/5) = 3/4

Therefore, cot Î¸ = 1/tan Î¸ = 4/3

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

**Question 2. If sin **Î¸** = 1/âˆš2, find all other trigonometric ratios of angle **Î¸.

**Solution:**

We are given, sin Î¸ = 1/âˆš2. So, cosec Î¸ =1/sin Î¸ = âˆš2.

Now we know,

=> cos Î¸ = âˆš(1 â€“ sin

^{2}Î¸)=> cos Î¸ = âˆš(1 â€“ (1/âˆš2)

^{2})=> cos Î¸ = âˆš(1 â€“ (1/2))

=> cos Î¸ = âˆš(1/2)

=> cos Î¸ = 1/âˆš2

So, sec Î¸ = 1/cos Î¸ = âˆš2

And tan Î¸ = sin Î¸/cos Î¸ = (1/âˆš2)/(1/âˆš2) = 1

Therefore, cot Î¸ = 1/tan Î¸ = 1

If sin Î¸ = 1/âˆš2, value of cosec Î¸, cos Î¸, sec Î¸, tan Î¸ and cot Î¸ are âˆš2, 1/âˆš2, âˆš2, 1 and 1 respectively.

**Question 3. If tan **Î¸** = 1/âˆš2, find the value of ****.**

**Solution:**

We are given, tan Î¸ = 1/âˆš2. Now we know,

=> sec Î¸ = âˆš(1 + tan

^{2}Î¸)=> sec Î¸ = âˆš(1 + (1/âˆš2)

^{2})=> sec Î¸ = âˆš(1+(1/2))

=> sec Î¸ = âˆš(3/2)

And cot Î¸ = 1/tan Î¸ = âˆš2. Also, we know,

=> cosec Î¸ = âˆš(1 + cot

^{2}Î¸)=> cosec Î¸ = âˆš(1 + (âˆš2)

^{2})=> cosec Î¸ = âˆš(1 + 2)

=> cosec Î¸ = âˆš3

So, =

=

=

Therefore, the value ofis.

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

**Solution:**

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

=> sec Î¸ = âˆš(1 + tan

^{2}Î¸)=> sec Î¸ = âˆš(1 + (3/4)

^{2})=> sec Î¸ = âˆš(1+(9/16))

=> sec Î¸ = âˆš(25/16)

=> sec Î¸ = 5/4

And cos Î¸ = 1/sec Î¸ = 4/5.

So, =

=

=

Therefore, the value ofis.

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

**Solution:**

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

Now we know,

=> cosec Î¸ = âˆš(1 + cot

^{2}Î¸)=> cosec Î¸ = âˆš(1 + (5/12)

^{2})=> cosec Î¸ = âˆš(1 + (25/144))

=> cosec Î¸ = âˆš(169/144)

=> cosec Î¸ = 13/12

And sin Î¸ = 1/cosec Î¸ = 12/13.

So, =

=

= 25

Therefore, the value ofis 25.

**Question 6. If cot **Î¸** = 1/âˆš3, find the value of ****.**

**Solution:**

We are given, cot Î¸ = 1/âˆš3. Now we know,

=> cosec Î¸ = âˆš(1 + cot

^{2}Î¸)=> cosec Î¸ = âˆš(1 + (1/âˆš3)

^{2})=> cosec Î¸ = âˆš(1 + (1/3))

=> cosec Î¸ = âˆš(4/3)

=> cosec Î¸ = 2/âˆš3

And sin Î¸ = 1/cosec Î¸ = âˆš3/2. Also, we know,

=> cos Î¸ = âˆš(1 â€“ sin

^{2}Î¸)=> cos Î¸ = âˆš(1 â€“ (âˆš3/2)

^{2})=> cos Î¸ = âˆš(1 â€“ (3/4))

=> cos Î¸ = âˆš(1/4)

=> cos Î¸ = 1/2

So, =

=

=

=

Therefore, the value ofis.

**Question 7. If cosec A = âˆš2, find the value of ****.**

**Solution:**

We are given, cosec A = âˆš2. So, sin A = 1/cosec A = 1/âˆš2.

Now we know,

=> cos A = âˆš(1 â€“ sin

^{2}A)=> cos A = âˆš(1 â€“ (1/âˆš2)

^{2})=> cos A = âˆš(1 â€“ (1/2))

=> cos A = âˆš(1/2)

=> cos A = 1/âˆš2

Hence, tan A = sin A/cos A = (1/âˆš2)/(1/âˆš2) = 1. And cot A = 1/tan A = 1.

So, =

=

= 2

Therefore, the value ofis 2.

**Question 8. If cot **Î¸** = âˆš3, find the value of ****.**

**Solution:**

We are given cot Î¸ = âˆš3. And tan Î¸ = 1/cot Î¸ = 1/âˆš3.

Now we know,

=> cosec Î¸ = âˆš(1 + cot

^{2}Î¸)=> cosec Î¸ = âˆš(1 + (âˆš3)

^{2})=> cosec Î¸ = âˆš4

=> cosec Î¸ = 2

Also, we know,

=> sec Î¸ = âˆš(1 + tan

^{2}Î¸)=> sec Î¸ = âˆš(1 + (1/âˆš3)

^{2})=> sec Î¸ = âˆš(1+(1/3))

=> sec Î¸ = âˆš(4/3)

=> sec Î¸ = 2/âˆš3

So, =

=

=

Therefore, the value ofis.

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

**Solution:**

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

=> sin Î¸ = âˆš(1 â€“ cos

^{2}Î¸)=> sin Î¸ = âˆš(1 â€“ (1/3)

^{2})=> sin Î¸ = âˆš(1 â€“ (1/9))

=> sin Î¸ = âˆš(8/9)

=> sin Î¸ = 2âˆš2/3

Hence, tan Î¸ = sin Î¸/cos Î¸ = (2âˆš2/3)/(1/3) = 2âˆš2

So, =

=

=

= 10

Therefore, the value ofis 10.

**Question 10. If âˆš3 tan **Î¸ = sin Î¸, find the value of sin^{2} Î¸ â€“ cos^{2} Î¸.** **

**Solution:**

We are given, âˆš3 tan Î¸ = sin Î¸

=> âˆš3 (sin Î¸/cos Î¸) = sin Î¸

=> cos Î¸ = 1/âˆš3

Now we know,

=> sin Î¸ = âˆš(1 â€“ cos

^{2}Î¸)=> sin Î¸ = âˆš(1 â€“ (1/âˆš3)

^{2})=> sin Î¸ = âˆš(1 â€“ (1/3))

=> sin Î¸ = âˆš(2/3)

So, sin

^{2}Î¸ â€“ cos^{2}Î¸ = âˆš(2/3)^{2}â€“ (1/âˆš3)^{2}= 2/3 â€“ 1/3

= 1/3

Therefore, the value of sin^{2}Î¸ â€“ cos^{2}Î¸ is 1/3.

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

**Solution:**

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

Now we know,

=> cos Î¸ = âˆš(1 â€“ sin

^{2}Î¸)=> cos Î¸ = âˆš(1 â€“ (12/13)

^{2})=> cos Î¸ = âˆš(1 â€“ (144/169))

=> cos Î¸ = âˆš(25/169)

=> cos Î¸ = 5/13

So, =

=

=

= 3

Therefore, the value ofis 3.

**Question 12. If sin **Î¸ + cos Î¸ = âˆš2 cos (90^{o}â€“Î¸), find cot Î¸.

**Solution:**

We are given,

=> sin Î¸ + cos Î¸ = âˆš2 cos (90

^{o}â€“Î¸)=> sin Î¸ + cos Î¸ = âˆš2 sin Î¸

=> cos Î¸ = (âˆš2â€“1)sin Î¸

=> cos Î¸/sin Î¸ = âˆš2â€“1

=> cot Î¸ = âˆš2â€“1

Therefore, value of cot Î¸ is âˆš2â€“1.

**Question 13. If 2sin**^{2} Î¸ â€“ cos^{2} Î¸ = 2, then find the value of Î¸.

^{2}

**Solution:**

We have,

=> 2sin

^{2}Î¸ â€“ cos^{2 }Î¸ = 2=> 2sin

^{2}Î¸ â€“ (1 â€“ sin^{2}Î¸) = 2=> 2sin

^{2}Î¸ â€“ 1 + sin^{2}Î¸ = 2=> 3sin

^{2}Î¸ = 3=> sin

^{2}Î¸ = 1=> sin Î¸ = 1

=> Î¸ = 90

^{o}

Therefore, the value of Î¸ is 90^{o}.

**Question 14. If âˆš3tan **Î¸ â€“ 1 = 0, find the value of sin^{2} Î¸ â€“ cos^{2} Î¸.

**Solution:**

We are given,

=> âˆš3tan Î¸ â€“ 1 = 0

=> tan Î¸ = 1/âˆš3

=> tan Î¸ = tan 30

^{o}=> Î¸ = 30

^{o}So, sin

^{2}Î¸ â€“ cos^{2}Î¸ = sin^{2}30^{o}â€“ cos^{2}30^{o}= (1/2)

^{2}â€“ (âˆš3/2)^{2}= (1/4) â€“ (3/4)

= â€“2/4

= â€“1/2

Therefore, the value of sin^{2}Î¸ â€“ cos^{2}Î¸ is â€“1/2.

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