Rd Sharma Xi 2018 Solutions for Class 11 Humanities Math Chapter 8 Transformation Formulae are provided here with simple step-by-step explanations. These solutions for Transformation Formulae are extremely popular among Class 11 Humanities students for Math Transformation Formulae Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2018 Book of Class 11 Humanities Math Chapter 8 are provided here for you for free. You will also love the ad-free experience on Meritnationâ€™s Rd Sharma Xi 2018 Solutions. All Rd Sharma Xi 2018 Solutions for class Class 11 Humanities Math are prepared by experts and are 100% accurate.

#### Question 1:

Express each of the following as the product of sines and cosines:
(i) sin 12x + sin 4x
(ii) sin 5x − sin x
(iii) cos 12x + cos 8x
(iv) cos 12x − cos 4x
(v) sin 2x + cos 4x

(i)

(ii)

(iii)

(iv)

(v)

#### Question 2:

Prove that:
(i) sin 38° + sin 22° = sin 82°
(ii) cos 100° + cos 20° = cos 40°
(iii) sin 50° + sin 10° = cos 20°
(iv) sin 23° + sin 37° = cos 7°
(v) sin 105° + cos 105° = cos 45°
(vi) sin 40° + sin 20° = cos 10°

(i)

(ii)

(iii)

(iv)

(v)

(vi)

#### Question 3:

Prove that:
(i) cos 55° + cos 65° + cos 175° = 0

(ii) sin 50° − sin 70° + sin 10° = 0

(iii) cos 80° + cos 40° − cos 20° = 0

(iv) cos 20° + cos 100° + cos 140° = 0

(v)

(vi) $\mathrm{cos}\frac{\mathrm{\pi }}{12}-\mathrm{sin}\frac{\mathrm{\pi }}{12}=\frac{1}{\sqrt{2}}$

(vii) sin 80° − cos 70° = cos 50°

(viii) sin 51° + cos 81° = cos 21°

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Prove that:
(i)

(ii)

(i)

(ii)

#### Question 5:

Prove that:
(i)
(ii) sin 47° + cos 77° = cos 17°

(i)

(ii)

#### Question 6:

Prove that:
(i) cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
(ii) cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
(iii) sin A + sin 2A + sin 4A + sin 5A = 4 cos $\frac{A}{2}$ cos $\frac{3A}{2}$ sin 3A
(iv) sin 3A + sin 2A − sin A = 4 sin A cos $\frac{A}{2}$ cos $\frac{3A}{2}$
(v) cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −$\frac{3}{4}$
(vi)
(vii)

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Hence, LHS = RHS

(vii)

Hence, LHS = RHS

Disclaimer: The given question is incorrect. The correct question should be .

Prove that:

(i)

(ii)

(iii)

(iv)

(v)

(i)

(ii)

(iii)

(iv)

(v)

Prove that:
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

#### Question 9:

Prove that:
(i)

(ii) cos (A + B + C) + cos (AB + C) + cos (A + BC) + cos (− A + B + C) = 4 cos A cos B cos C

#### Question 10:

Given:
sin A + sin B = $\frac{1}{4}$         .....(i)
cos A + cos B =$\frac{1}{2}$         .....(ii)

Dividing (i) by (ii):

#### Question 11:

If cosec A + sec A = cosec B + sec B, prove that tan A tan B = $\mathrm{cot}\frac{A+B}{2}$.

Given:

#### Question 12:

.

Given:
sin 2A = λ sin 2B

$⇒\frac{\mathrm{sin}2A}{\mathrm{sin}2B}=\lambda$

$⇒\frac{\mathrm{tan}\left(A+B\right)}{\mathrm{tan}\left(A-B\right)}=\frac{\lambda +1}{\lambda -1}\phantom{\rule{0ex}{0ex}}$

Hence proved.

#### Question 13:

Prove that:
(i)

(ii) sin (B C) cos (AD) + sin (CA) cos (BD) + sin (AB) cos (CD) = 0

#### Question 15:

If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ

#### Question 16:

If y sin Ï• = x sin (2θ + Ï•), prove that (x + y) cot (θ + Ï•) = (yx) cot θ.

Given:
y sin Ï• = x sin (2θ + Ï•)

#### Question 17:

If cos (A + B) sin (CD) = cos (AB) sin (C + D), prove that tan A tan B tan C + tan D = 0.

cos (A + B) sin (CD) = cos (AB) sin (C + D)

$⇒$[cosA cosB − sinA sinB] [sinC cosD − cosC sinD] = [cosA cosB + sinA sinB] [sinC cosD +  cosC sinD]

#### Question 18:

If , prove that $xy+yz+zx=0$.                    [NCERT EXEMPLAR]

#### Question 19:

If , prove that .                           [NCERT EXEMPLAR]

Given:

$⇒\frac{m}{n}=\frac{\mathrm{sin}\left(\theta +2\alpha \right)}{\mathrm{sin}\theta }$

Applying componendo and dividendo, we get

#### Question 1:

If (cos α + cos β)2 + (sin α + sin β)2, write the value of λ.

(cos α + cos β)2 + (sin α + sin β)2

Consider LHS:
(cos α + cos β)2 + (sin α + sin β)2

#### Question 2:

Write the value of sin $\frac{\mathrm{\pi }}{12}$ sin $\frac{5\mathrm{\pi }}{12}$.

sin $\frac{\mathrm{\pi }}{12}$ sin $\frac{5\mathrm{\pi }}{12}$

#### Question 3:

If sin A + sin B = α and cos A + cos B = β, then write the value of tan $\left(\frac{A+B}{2}\right)$.

Given:
sin A + sin B = α            .....(i)
cos A + cos B = β           .....(ii)
Dividing (i) by (ii):

#### Question 4:

If cos A = m cos B, then write the value of .

#### Question 5:

Write the value of the expression .

#### Question 6:

If A + B = $\frac{\mathrm{\pi }}{3}$ and cos A + cos B = 1, then find the value of cos $\frac{A-B}{2}$.

Given:
A + B = $\frac{\mathrm{\pi }}{3}$
and cos A + cos B = 1

#### Question 7:

Write the value of $\mathrm{sin}\frac{\mathrm{\pi }}{15}\mathrm{sin}\frac{4\mathrm{\pi }}{15}\mathrm{sin}\frac{3\mathrm{\pi }}{10}$

#### Question 8:

If sin 2A = λ sin 2B, then write the value of $\frac{\lambda +1}{\lambda -1}$.

Given:
sin 2A = λ sin 2B

$⇒\frac{\mathrm{sin}2A}{\mathrm{sin}2B}=\lambda$

#### Question 9:

Write the value of .

#### Question 10:

If cos (A + B) sin (CD) = cos (AB) sin (C + D), then write the value of tan A tan B tan C.

cos (A + B) sin (CD) = cos (AB) sin (C + D)

$⇒$[cosA cosB − sinA sinB] [sinC cosD − cosC sinD] =  [cosA cosB + sinA sinB] [sinC cosD + cosC sinD]

#### Question 1:

cos 40° + cos 80° + cos 160° + cos 240° =
(a) 0

(b) 1

(c) $\frac{1}{2}$

(d) $-\frac{1}{2}$

(d) $-\frac{1}{2}$

#### Question 2:

sin 163° cos 347° + sin 73° sin 167° =
(a) 0

(b) $\frac{1}{2}$

(c) 1

(d) None of these

(b) $\frac{1}{2}$

#### Question 3:

If sin 2 θ + sin 2 Ï• = $\frac{1}{2}$ and cos 2 θ + cos 2 Ï• = $\frac{3}{2}$, then cos2 (θ − Ï•) =
(a) $\frac{3}{8}$

(b) $\frac{5}{8}$

(c) $\frac{3}{4}$

(d) $\frac{5}{4}$

(b) $\frac{5}{8}$
Given:
sin 2θ + sin 2Ï• = $\frac{1}{2}$                  .....(i)
and
cos 2θ + cos 2Ï• = $\frac{3}{2}$         .....(ii)

Squaring and adding (i) and (ii), we get:
(sin 2θ + sin 2Ï•)2 + (cos 2θ + cos 2Ï•)2 = $\frac{1}{4}+\frac{9}{4}$

#### Question 4:

The value of cos 52° + cos 68° + cos 172° is
(a) 0
(b) 1
(c) 2
(d) 3/2

(a) 0

#### Question 5:

The value of sin 78° − sin 66° − sin 42° + sin 60° is
(a) $\frac{1}{2}$

(b) $-\frac{1}{2}$

(c) −1

(d) None of these

(d) None of these

#### Question 6:

If sin α + sin β = a and cos α − cos β = b, then tan $\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}$=
(a) $-\frac{a}{b}$

(b) $-\frac{b}{a}$

(c) $\sqrt{{a}^{2}+{b}^{2}}\phantom{\rule{0ex}{0ex}}$

(d) None of these

(b) $-\frac{b}{a}$

Given:
sin α + sin β = a                  .....(i)
cos α − cos β = b                .....(ii)

Dividing (i) by (ii):

#### Question 7:

cos 35° + cos 85° + cos 155° =
(a) 0

(b) $\frac{1}{\sqrt{3}}$

(c) $\frac{1}{\sqrt{2}}$

(d) cos 275°

(a) 0

#### Question 8:

The value of sin 50° − sin 70° + sin 10° is equal to
(a) 1
(b) 0
(c) 1/2
(d) 2

(b) 0

#### Question 9:

sin 47° + sin 61° − sin 11° − sin 25° is equal to
(a) sin 36°
(b) cos 36°
(c) sin 7°
(d) cos 7°

(d) cos 7°

#### Question 10:

If cos A = m cos B, then =
(a) $\frac{m-1}{m+1}$

(b) $\frac{m+2}{m-2}$

(c)$\frac{m+1}{m-1}$

(d) None of these

(c)$\frac{m+1}{m-1}$

#### Question 11:

If A, B, C are in A.P., then =
(a) tan B
(b) cot B
(c) tan 2 B
(d) None of these

(b) cot B

Since A,B and C are in A.P,

$=\frac{\mathrm{sin}\left(\frac{A-C}{2}\right)\mathrm{cos}\left(\frac{A+C}{2}\right)}{\mathrm{sin}\left(\frac{A+C}{2}\right)\mathrm{sin}\left(\frac{A-C}{2}\right)}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{cos}\left(\frac{A+C}{2}\right)}{\mathrm{sin}\left(\frac{A+C}{2}\right)}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{cos}B}{\mathrm{sin}B}\phantom{\rule{0ex}{0ex}}=\mathrm{cot}B$

#### Question 12:

If sin (B + CA), sin (C + AB), sin (A + BC) are in A.P., then cot A, cot B and cot C are in
(a) GP
(b) HP
(c) AP
(d) None of these

(b) HP

Given:
sin (B + CA), sin (C + AB) and sin (A + BC) are in A.P.

Hence, cotA, cotB and cotC are in HP.

#### Question 13:

If sin x + sin y = $\sqrt{3}$ (cos y − cos x), then sin 3x + sin 3y =
(a) 2 sin 3x
(b) 0
(c) 1
(d) none of these

We have,
sin x + sin y = $\sqrt{3}$ (cos y − cos x)

$\mathrm{Case}-\mathrm{I}$

${\mathrm{sin}}3x+\mathrm{sin}3y=\mathrm{sin}\left(-3y\right)+\mathrm{sin}3y=-\mathrm{sin}3y+\mathrm{sin}3y=0$

#### Question 14:

If $\mathrm{tan}\alpha =\frac{x}{x+1}$ and $\mathrm{tan}\beta =\frac{1}{2x+1}$, then $\alpha +\beta$ is equal to

(a) $\frac{\mathrm{\pi }}{2}$                                  (a) $\frac{\mathrm{\pi }}{3}$                                  (a) $\frac{\mathrm{\pi }}{6}$                                  (a) $\frac{\mathrm{\pi }}{4}$

It is given that $\mathrm{tan}\alpha =\frac{x}{x+1}$ and $\mathrm{tan}\beta =\frac{1}{2x+1}$.

Now,

$=\frac{2{x}^{2}+2x+1}{2{x}^{2}+2x+1}\phantom{\rule{0ex}{0ex}}=1$

$\therefore \mathrm{tan}\left(\alpha +\beta \right)=1=\mathrm{tan}\frac{\mathrm{\pi }}{4}\phantom{\rule{0ex}{0ex}}⇒\alpha +\beta =\frac{\mathrm{\pi }}{4}$

Hence, the correct answer is option D.

#### Question 1:

Express each of the following as the sum or difference of sines and cosines:
(i) 2 sin 3x cos x
(ii) 2 cos 3x sin 2x
(iii) 2 sin 4x sin 3x
(iv) 2 cos 7x cos 3x

(i)

(ii)

(iii)

(iv)

#### Question 2:

Prove that:
(i) $2\mathrm{sin}\frac{5\mathrm{\pi }}{12}\mathrm{sin}\frac{\mathrm{\pi }}{12}=\frac{1}{2}$

(ii) $2\mathrm{cos}\frac{5\mathrm{\pi }}{12}\mathrm{cos}\frac{\mathrm{\pi }}{12}=\frac{1}{2}$

(iii) $2\mathrm{sin}\frac{5\mathrm{\pi }}{12}\mathrm{cos}\frac{\mathrm{\pi }}{12}=\frac{\sqrt{3}+2}{2}$

(i)

(ii)

(iii)

Show that :
(i)

(ii)

(i)

(ii)

#### Question 5:

Prove that:
(i) cos 10° cos 30° cos 50° cos 70° = $\frac{3}{16}$

(ii) cos 40° cos 80° cos 160° = $-\frac{1}{8}$

(iii) sin 20° sin 40° sin 80° = $\frac{\sqrt{3}}{8}$

(iv) cos 20° cos 40° cos 80° = $\frac{1}{8}$

(v) tan 20° tan 40° tan 60° tan 80° = 3

(vi) tan 20° tan 30° tan 40° tan 80° = 1

(vii) sin 10° sin 50° sin 60° sin 70° = $\frac{\sqrt{3}}{16}$

(viii) sin 20° sin 40° sin 60° sin 80° = $\frac{3}{16}$

(i)

(ii)

(iii)

(iv)

(v)
LHS = tan 20° tan 40° tan 60° tan 80°

(vi)
LHS = tan 20° tan 30° tan 60° tan 80°

(vii)

(viii)

#### Question 6:

Show that:
(i) sin A sin (BC) + sin B sin (CA) + sin C sin (AB) = 0
(ii) sin (BC) cos (AD) + sin (CA) cos (BD) + sin (AB) cos (CD) = 0

(i)

(ii)

#### Question 7:

Prove that

$\frac{\pi }{3}=60°$

$=\frac{\mathrm{sin}x\left({\mathrm{sin}}^{2}60°-{\mathrm{sin}}^{2}x\right)}{\mathrm{cos}x\left({\mathrm{cos}}^{2}60°-{\mathrm{sin}}^{2}x\right)}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{sin}x\left(\frac{3}{4}-{\mathrm{sin}}^{2}x\right)}{\mathrm{cos}x\left(\frac{1}{4}-{\mathrm{sin}}^{2}x\right)}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{sin}x\left(3-4{\mathrm{sin}}^{2}x\right)}{\mathrm{cos}x\left(1-4{\mathrm{sin}}^{2}x\right)}$

$=\frac{3\mathrm{sin}x-4{\mathrm{sin}}^{3}x}{4{\mathrm{cos}}^{3}x-3\mathrm{cos}x}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{sin}3x}{\mathrm{cos}3x}\phantom{\rule{0ex}{0ex}}=\mathrm{tan}3x=\mathrm{RHS}$

#### Question 8:

If α + β = $\frac{\pi }{2}$, show that the maximum value of cos α cos β is $\frac{1}{2}$.

$\frac{\pi }{2}=90°$