## Selina Concise Mathematics Class 10 ICSE Solutions Chapter 21 Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables) Ex 21B

These Solutions are part of Selina Concise Mathematics Class 10 ICSE Solutions. Here we have given Selina Concise Mathematics Class 10 ICSE Solutions Chapter 21 Trigonometrical Identities Ex 21B.

**Other Exercises**

- Selina Concise Mathematics Class 10 ICSE Solutions Chapter 21 Trigonometrical Identities Ex 21A
- Selina Concise Mathematics Class 10 ICSE Solutions Chapter 21 Trigonometrical Identities Ex 21B
- Selina Concise Mathematics Class 10 ICSE Solutions Chapter 21 Trigonometrical Identities Ex 21C
- Selina Concise Mathematics Class 10 ICSE Solutions Chapter 21 Trigonometrical Identities Ex 21D
- Selina Concise Mathematics Class 10 ICSE Solutions Chapter 21 Trigonometrical Identities Ex 21E

**Question 1.
**

**Prove that:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Question 2.
**

**If xcosA + ysinA = m and xsinA-ycosA = n, then prove that: x**

^{2}+y^{2}= m^{2}+ n^{2 }**Solution:**

x cos A + y sin A = m …(i)

x sin A – y cos A = n ….(ii)

squaring (i) and (ii)

x

^{2}cos

^{2}A + y

^{2}sin

^{2}A + 2 xy cosA sinA = m

^{2 }x

^{2}sin

^{2}A + y

^{2}cos

^{2}A – 2 xy cos A sin A = n

^{2 }Adding we get,

x

^{2}(sin

^{2}A + cos

^{2}A) + y

^{2}(sin

^{2}A + cos

^{2}A) = m

^{2}+n

^{2 }∴ x

^{2}+y

^{2}= m

^{2}+ n

^{2}(∵ sin

^{2}A + cos

^{2}A= 1)

Hence proved.

**Question 3.
**

**If m = a sec A +b tan A and n=atanA + bsecA, then prove that: m**

^{2}-n^{2}= a^{2}-b^{2 }

**Solution:**

m = asec A + btan A ……(i)

n = a tan A + b sec A …..(ii)

squaring (i) and (ii)

m

^{2}= a

^{2}sec

^{2}A + b

^{2}tan

^{2}A + 2ab sec A tan A

n

^{2}= a

^{2}tan

^{2}A + b

^{2}sec

^{2}A + 2 ab tan A sec A

Subtracting, we get

m

^{2}– n

^{2}= a

^{2}(sec

^{2}A – tan

^{2}A) + b

^{2}(tan

^{2}A – sec

^{2}A)

= a

^{2}x 1 +b

^{2}(-1) = a

^{2}-b

^{2 }( ∵ sec

^{2}A-tan

^{2}A= 1) .

Henceproved

**Question 4.
**

**If x = r sin A cos B, y = r sin A sin B and z = r cos A, then prove that: x**

^{2}+ y^{2}+ z^{2}= r.**Solution:**

x = r sin A cos B ….(i)

y = r sin A sin B ….(ii)

z = r cosA …….(iii)

Squaring, (i), (ii) & (iii)

x

^{2}=r

^{2}sin

^{2}A cos

^{2}B,

y

^{2}= r

^{2}sin

^{2}Asin

^{2}B,

z

^{2}= r

^{2}cos

^{2}A

Adding, we get,

x

^{2}+y

^{2}+ z

^{2}=r

^{2}(sin

^{2}A cos

^{2}E + sin

^{2}A sin

^{2}B+cos

^{2}A)

= r[sin

^{2}A (cos

^{2}B + sin

^{2}B) + cos

^{2}A]

= r [sin

^{2}A x 1 + cos

^{2}A]

= r

^{2}[sin

^{2}A + cos

^{2}A] = r

^{2}x 1 = r

^{2}( ∵ sin

^{2}A + cos

^{2}A = 1)

Hence proved.

**Question 5.
**

**If sin A + cos A = m and sec A + cosec A=n, show that n (m**

^{2}-1) = 2m**Solution:**

**Question 6.
**

**If x = r cos A cos B, y = r cos A sin B and z = r sin A, show that x**

^{2}+ y^{2}+ z^{2}= r^{2 }

**Solution:**

x = r cosAcosB ….(i)

y = r cosAsinB ….(ii)

z = r sinA ….(iii)

Squaring (i), (ii), (iii)

x

^{2}= r

^{2}cos

^{2}A cos

^{2}B, y

^{2}= r

^{2}cos

^{2}A sin

^{2}B

z

^{2}= r

^{2}sin

^{2}A

Adding, we get

x

^{2}+ y

^{2}+ z

^{2}= r

^{2}(cos

^{2}A cos

^{2}B + cos

^{2}A sin

^{2}B + sin

^{2}A)

= r

^{2}[cos

^{2}A (cos

^{2}B + sin

^{2}B) + sin

^{2}A]

= r

^{2}[cos

^{2}Ax 1+sin

^{2}A]

= r

^{2}(1) = r

^{2 `}Hence proved.

**Question 7.**

**Solution:**

**P.Q.**

**Solution:**

**P.Q.
**

**Evaluate:**

**Solution:**

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