Rd Sharma Xi 2019 Solutions for Class 11 Humanities Math Chapter 7 Values Of Trigonometric Functions At Sum Or Difference Of Angles are provided here with simple step-by-step explanations. These solutions for Values Of Trigonometric Functions At Sum Or Difference Of Angles are extremely popular among Class 11 Humanities students for Math Values Of Trigonometric Functions At Sum Or Difference Of Angles Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2019 Book of Class 11 Humanities Math Chapter 7 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2019 Solutions. All Rd Sharma Xi 2019 Solutions for class Class 11 Humanities Math are prepared by experts and are 100% accurate.

#### Question 1:

If and , where 0 < A, $\mathrm{B}<\frac{\mathrm{\pi }}{2}$, find the values of the following:

(i) sin (A + B)
(ii) cos (A + B)
(iii) sin (A − B)
(iv) cos (A − B)

Now,

#### Question 2:

(a) If , where $\frac{\mathrm{\pi }}{2}$< A < π and 0 < B < $\frac{\mathrm{\pi }}{2}$, find the following:

(i) sin (A + B)
(ii) cos (A + B)

(b) If , where A and B both lie in second quadrant, find the value of sin (A + B).

#### Question 3:

If , where π < A < $\frac{3\mathrm{\pi }}{2}\mathrm{and}\frac{3\mathrm{\pi }}{2}$< B < 2π, find the following:

(i) sin (A + B)
(ii) cos (A + B)

#### Question 4:

If , where π < A < $\frac{3\mathrm{\pi }}{2}$and 0 < B < $\frac{\mathrm{\pi }}{2}$, find tan (A + B).

#### Question 5:

If , where $\frac{\mathrm{\pi }}{2}$< A < π and $\frac{3\mathrm{\pi }}{2}$ < B < 2π, find tan (AB).

#### Question 6:

If , where $\frac{\mathrm{\pi }}{2}$ < A < π and 0 < B < $\frac{\mathrm{\pi }}{2}$, find the following:

(i) tan (A + B)
(ii) tan (AB)

#### Question 7:

Evaluate the following:

(i) sin 78° cos 18° − cos 78° sin 18°
(ii) cos 47° cos 13° − sin 47° sin 13°
(iii) sin 36° cos 9° + cos 36° sin 9°
(iv) cos 80° cos 20° + sin 80° sin 20°

#### Question 8:

If , where A lies in the second quadrant and B in the third quadrant, find the values of the following:

(i) sin (A + B)
(ii) cos (A + B)
(iii) tan (A + B)

#### Question 9:

Prove that: $\frac{7\pi }{12}+\mathrm{cos}\frac{\pi }{12}=\mathrm{sin}\frac{5\pi }{12}-\mathrm{sin}\frac{\pi }{12}$

LHS = cos105o + cos15o
= cos(90o + 15o) + cos(90o$-$ 75o)
= - sin 15o + sin 75o                      [As cos(90o+A) = $-$ sin A and cos(90o$-$B) = sin B]
= sin 75o$-$ sin 15o
= RHS
Hence proved.

Prove that .

Prove that
(i) .

(ii)
(ii)

(i)

#### Question 12:

Prove that:
(i) $\mathrm{sin}\left(\frac{\pi }{3}-x\right)\mathrm{cos}\left(\frac{\pi }{6}+x\right)+\mathrm{cos}\left(\frac{\pi }{3}-x\right)\mathrm{sin}\left(\frac{\pi }{6}+x\right)=1$
(ii) $\mathrm{sin}\left(\frac{4\mathrm{\pi }}{9}+7\right)\mathrm{cos}\left(\frac{\mathrm{\pi }}{9}+7\right)-\mathrm{cos}\left(\frac{4\mathrm{\pi }}{9}+7\right)\mathrm{sin}\left(\frac{\mathrm{\pi }}{9}+7\right)=\frac{\sqrt{3}}{2}$
(iii) $\mathrm{sin}\left(\frac{3\mathrm{\pi }}{8}-5\right)\mathrm{cos}\left(\frac{\mathrm{\pi }}{8}+5\right)+\mathrm{cos}\left(\frac{3\mathrm{\pi }}{8}-5\right)\mathrm{sin}\left(\frac{\mathrm{\pi }}{8}+5\right)=1$

(i)

(ii)

(iii)

Prove that .

#### Question 14:

(i) If , prove that $A+B=\frac{\mathrm{\pi }}{4}$.
(ii) If , then prove that $A-B=\frac{\mathrm{\pi }}{4}$.

(i)

(ii)

#### Question 15:

Prove that:
(i) ${\mathrm{cos}}^{2}45°-{\mathrm{sin}}^{2}15°=\frac{\sqrt{3}}{4}$
(ii) sin2 (n + 1) A − sin2nA = sin (2n + 1) A sin A.

(i)

Hence proved.

(ii)

#### Question 16:

Prove that:

(i)
(ii)
(iii)
(iv) sin2B = sin2A + sin2 (AB) − 2 sin A cos B sin (A B)
(v) cos2A + cos2B − 2 cos A cos B cos (A + B) = sin2 (A + B)
(vi)

#### Question 17:

Prove that:
(i) tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
(ii) $\mathrm{tan}\frac{\pi }{12}+\mathrm{tan}\frac{\pi }{6}+\mathrm{tan}\frac{\pi }{12}\mathrm{tan}\frac{\pi }{6}=1$
(iii) tan 36° + tan 9° + tan 36° tan 9° = 1
(iv) tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x

Prove that:

#### Question 19:

Prove that sin2 (n + 1) A − sin2nA = sin (2n + 1) A sin A.

#### Question 20:

If tan A = x tan B, prove that .

#### Question 21:

If tan (A + B) = x and tan (AB) = y, find the values of tan 2A and tan 2B.

#### Question 22:

If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.

#### Question 23:

If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) $\frac{1}{a}-\frac{1}{b}$.

Given:

#### Question 24:

If x lies in the first quadrant and , then prove that:

#### Question 25:

If tan x + , then prove that .

#### Question 26:

If sin (α + β) = 1 and sin (α − β)$=\frac{1}{2}$, where 0 ≤ α, $\mathrm{\beta }\le \frac{\mathrm{\pi }}{2}$, then find the values of tan (α + 2β) and tan (2α + β).

#### Question 27:

If α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β).

#### Question 28:

If sin α + sin β = a and cos α + cos β = b, show that

(i)
(ii)

(i)

Now,

From (1) and (2), we have

(ii)

Prove that:

(i)
(ii)
(iii)

#### Question 30:

If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.

Given:

#### Question 31:

If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.

#### Question 32:

If angle $\theta$ is divided into two parts such that the tangents of one part is $\lambda$ times the tangent of other, and $\varphi$ is their difference, then show that $\mathrm{sin}\theta =\frac{\lambda +1}{\lambda -1}\mathrm{sin}\varphi$.                                                                                                                                                          [NCERT EXEMPLER]

Let $\alpha$ and $\beta$ be the two parts of angle $\theta$. Then,

$\theta =\alpha +\beta$ and $\varphi =\alpha -\beta$               (Given)

Now,

Applying componendo and dividendo, we get

#### Question 33:

If $\mathrm{tan}\theta =\frac{\mathrm{sin}\alpha -\mathrm{cos}\alpha }{\mathrm{sin}\alpha +\mathrm{cos}\alpha }$, then show that $\mathrm{sin}\alpha +\mathrm{cos}\alpha =\sqrt{2}\mathrm{cos}\theta$.                              [NCERT EXEMPLER]

$\mathrm{tan}\theta =\frac{\mathrm{sin}\alpha -\mathrm{cos}\alpha }{\mathrm{sin}\alpha +\mathrm{cos}\alpha }$

Dividing numerator and denominator on the RHS by $\mathrm{cos}\alpha$, we get

Now,

$\mathrm{sin}\alpha +\mathrm{cos}\alpha \phantom{\rule{0ex}{0ex}}=\mathrm{sin}\left(\frac{\mathrm{\pi }}{4}+\theta \right)+\mathrm{cos}\left(\frac{\mathrm{\pi }}{4}+\theta \right)\phantom{\rule{0ex}{0ex}}=\mathrm{sin}\frac{\mathrm{\pi }}{4}\mathrm{cos}\theta +\mathrm{cos}\frac{\mathrm{\pi }}{4}\mathrm{sin}\theta +\mathrm{cos}\frac{\mathrm{\pi }}{4}\mathrm{cos}\theta -\mathrm{sin}\frac{\mathrm{\pi }}{4}\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}=\frac{1}{\sqrt{2}}\mathrm{cos}\theta +\frac{1}{\sqrt{2}}\mathrm{sin}\theta +\frac{1}{\sqrt{2}}\mathrm{cos}\theta -\frac{1}{\sqrt{2}}\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}=\frac{2}{\sqrt{2}}\mathrm{cos}\theta \phantom{\rule{0ex}{0ex}}=\sqrt{2}\mathrm{cos}\theta$

#### Question 34:

If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).

#### Question 1:

Find the maximum and minimum values of each of the following trigonometrical expressions:
(i) 12 sin x − 5 cos x
(ii) 12 cos x + 5 sin x + 4
(iii)
(iv) sin x − cos x + 1

(i)

(ii)

(iii)

(iv)

#### Question 2:

Reduce each of the following expressions to the sine and cosine of a single expression:
(i)
(ii) cos x − sin x
(iii) 24 cos x + 7 sin x

#### Question 3:

Show that sin 100° − sin 10° is positive.

#### Question 4:

Prove that lies between .

#### Question 1:

If α + β − γ = π and sin2 α +sin2 β − sin2 γ = λ sin α sin β cos γ, then write the value of λ.

#### Question 2:

If x cos θ = y cos , then write the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$.

#### Question 3:

Write the maximum and minimum values of 3 cos x + 4 sin x + 5.

#### Question 4:

Write the maximum value of 12 sin x − 9 sin2x.

#### Question 5:

If 12 sin x − 9sin2x attains its maximum value at x = α, then write the value of sin α.

#### Question 6:

Write the interval in which the value of 5 cos x + 3 cos $\left(x+\frac{\pi }{3}\right)+3$ lies.

#### Question 7:

If tan (A + B) = p and tan (AB) = q, then write the value of tan 2B.

#### Question 8:

If , then write the value of tan x tan y.

#### Question 9:

If a = b , then write the value of ab + bc + ca.

#### Question 10:

If A + B = C, then write the value of tan A tan B tan C.

#### Question 11:

If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).

#### Question 12:

If tan $\mathrm{\alpha }=\frac{1}{1+{2}^{-x}}$and , then write the value of α + β lying in the interval .

#### Question 1:

The value of ${\mathrm{sin}}^{2}\frac{5\mathrm{\pi }}{12}-{\mathrm{sin}}^{2}\frac{\mathrm{\pi }}{12}$ is
(a) $\frac{1}{2}$
(b) $\frac{\sqrt{3}}{2}$
(c) 1
(d) 0

(b)  $\frac{\sqrt{3}}{2}$

#### Question 2:

If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
(a) 0
(b) −1
(c) 1
(d) None of these

(b)  −1
π = 180°
$\mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{\mathrm{cos}B\mathrm{cos}\left(\mathrm{\pi }-\left(A+B\right)\right)-\mathrm{sin}B\mathrm{sin}\left(\mathrm{\pi }-\left(A+B\right)\right)}{\mathrm{cos}A}$

We know that, ,

$\therefore \mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{\mathrm{cos}B\mathrm{cos}\left(A+B\right)-\mathrm{sin}B\mathrm{sin}\left(A+B\right)}{\mathrm{cos}A}$

Now, using the identities $\mathrm{cos}\left(A+B\right)=\mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B$ and $\mathrm{sin}\left(A+B\right)=\mathrm{sin}A\mathrm{cos}B+\mathrm{cos}A\mathrm{sin}B$, we get

$\mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{-\mathrm{cos}A\mathrm{cos}{B}^{2}+\mathrm{cos}B\mathrm{sin}A\mathrm{sin}B-\mathrm{sin}B\mathrm{sin}A\mathrm{cos}B-{\mathrm{sin}}^{2}B\mathrm{cos}A}{\mathrm{cos}A}$

$⇒\mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{-\mathrm{cos}A\left({\mathrm{cos}}^{2}B+{\mathrm{sin}}^{2}B\right)}{\mathrm{cos}A}\phantom{\rule{0ex}{0ex}}⇒\mathrm{sec}A\left(\mathrm{cos}B\mathrm{cos}C-\mathrm{sin}B\mathrm{sin}C\right)=\frac{-\mathrm{cos}A}{\mathrm{cos}A}=-1$

#### Question 3:

tan 20° + tan 40° + $\sqrt{3}$ tan 20° tan 40° is equal to

(a) $\frac{\sqrt{3}}{4}$
(b) $\frac{\sqrt{3}}{2}$
(c) $\sqrt{3}$
(d) 1

(c) $\sqrt{3}$

#### Question 4:

If , then the value of A + B is

(a) 0
(b) $\frac{\mathrm{\pi }}{2}$
(c) $\frac{\mathrm{\pi }}{3}$
(d) $\frac{\mathrm{\pi }}{4}$

(d)  $\frac{\mathrm{\pi }}{4}$

#### Question 5:

If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
(a) 0
(b) 5
(c) 1
(d) None of these

(a) 0

#### Question 6:

If in âˆ†ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =

(a) 6
(b) 1
(c) $\frac{1}{6}$
(d) None of these

(c) $\frac{1}{6}$
In triangle ABC,

If tan A+tan B+tan C =6,
tan A tan B tan C =6

#### Question 7:

tan 3A − tan 2A − tan A =

(a) tan 3 A tan 2 A tan A
(b) −tan 3 A tan 2 A tan A
(c) tan A tan 2 A − tan 2 A tan 3 A − tan 3 A tan A
(d) None of these

(a)

#### Question 8:

If A + B + C = π, then is equal to
(a) tan A tan B tan C
(b) 0
(c) 1
(d) None of these

(c) 1
π = 180°
Using tan(180 – A) = -tan A, we get:

#### Question 9:

If , where P and Q both are acute angles. Then, the value of PQ is
(a) $\frac{\mathrm{\pi }}{6}$

(b) $\frac{\mathrm{\pi }}{3}$

(c) $\frac{\mathrm{\pi }}{4}$

(d) $\frac{\mathrm{\pi }}{12}$

(b) 60â° = $\frac{\mathrm{\pi }}{3}$

$=\frac{1}{7}×\frac{13}{14}+\frac{4\sqrt{3}}{7}×\frac{3\sqrt{3}}{14}\phantom{\rule{0ex}{0ex}}=\frac{13+36}{98}\phantom{\rule{0ex}{0ex}}$
$=\frac{49}{98}$

Hence, the correct answer is option B.

#### Question 10:

If cot (α + β) = 0, sin (α + 2β) is equal to

(a) sin α
(b) cos 2 β
(c) cos α
(d) sin 2 α

(a)

(a) tan 55°
(b) cot 55°
(c) −tan 35°
(d) −cot 35°

(a)

#### Question 12:

The value of is
(a)
(b) 0
(c)
(d) $\frac{1}{2}$

(a)

#### Question 13:

If tan θ1 tan θ2 = k, then

(a) $\frac{1+k}{1-k}$
(b) $\frac{1-k}{1+k}$
(c) $\frac{k+1}{k-1}$
(d) $\frac{k-1}{k+1}$

(a) $\frac{1+k}{1-k}$

$\frac{1+\mathrm{tan}{\theta }_{1}\mathrm{tan}{\theta }_{2}}{1-\mathrm{tan}{\theta }_{1}\mathrm{tan}{\theta }_{2}}\phantom{\rule{0ex}{0ex}}=\frac{1+k}{1-k}$

#### Question 14:

If sin (π cos x) = cos (π sin x), then sin 2 x =

(a) $±\frac{3}{4}$

(b) $±\frac{4}{3}$

(c) $±\frac{1}{3}$

(d) none of these

#### Question 15:

If $\mathrm{tan}\theta =\frac{1}{2}$ and $\mathrm{tan}\varphi =\frac{1}{3}$, then the value of $\theta +\varphi$ is

(a) $\frac{\mathrm{\pi }}{6}$                               (b) $\mathrm{\pi }$                               (c) 0                               (d) $\frac{\mathrm{\pi }}{4}$

It is given that $\mathrm{tan}\theta =\frac{1}{2}$ and $\mathrm{tan}\varphi =\frac{1}{3}$.

Now,

Hence, the correct answer is option D.

#### Question 16:

The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is

(a) sin 2A
(b) cos 2A
(c) cos 3A
(d) sin 3A

(b)  cos 2A

#### Question 17:

If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
(a) a2 + 1
(b) a2 + 2
(c) a2 − 2
(d) None of these

(c) ${a}^{2}-2$

#### Question 18:

If tan (AB) = 1 and sec (A + B) = $\frac{2}{\sqrt{3}}$, the smallest positive value of B is

(a)
(b)
(c) $\frac{13\mathrm{\pi }}{24}$
(d)

(b)

#### Question 19:

If AB = π/4, then (1 + tan A) (1 − tan B) is equal to

(a) 2
(b) 1
(c) 0
(d) 3

(a)  $2$

#### Question 20:

The maximum value of ${\mathrm{sin}}^{2}\left(\frac{2\pi }{3}+x\right)+{\mathrm{sin}}^{2}\left(\frac{2\pi }{3}-x\right)$ is
(a) 1/2
(b) 3/2
(c) 1/4
(d) 3/4

(b) $\frac{3}{2}$
$\frac{2\pi }{3}=120°$

#### Question 21:

If cos (AB)$=\frac{3}{5}$and tan A tan B = 2, then

(a)
(b)
(c)
(d)

(a) $\frac{1}{5}$

#### Question 22:

If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =

(a) −1
(b) $\frac{1}{2}$
(c) $-\frac{1}{2}$
(d) None of these

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

#### Question 23:

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

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

It is given that $\mathrm{tan}\alpha =\frac{x}{x+1}$ and $\mathrm{tan}\beta =\frac{1}{2x+1}$.
$=\frac{2{x}^{2}+2x+1}{2{x}^{2}+2x+1}\phantom{\rule{0ex}{0ex}}=1$