RD Sharma solutions for Mathematics [English] Class 11 chapter 7 - Values of Trigonometric function at sum or difference of angles [Latest edition]

If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:

If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:

cos (A + B)

If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
sin (A − B)

If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A − B)

 If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
sin (A + B)

 If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
cos (A + B)

If \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], where A and B both lie in second quadrant, find the value of sin (A + B).

If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
sin (A + B)

If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
cos (A + B)

If \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (A + B).

If \[\sin A = \frac{1}{2}, \cos B = \frac{12}{13}\], where \[\frac{\pi}{2}\]< A < π and \[\frac{3\pi}{2}\] < B < 2π, find tan (A − B).

If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A + B)

If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A - B)

Evaluate the following:
sin 78° cos 18° − cos 78° sin 18°

Evaluate the following:
cos 47° cos 13° − sin 47° sin 13°

Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°

Evaluate the following:
 cos 80° cos 20° + sin 80° sin 20°

If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
sin (A + B)

If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
cos (A + B)

If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
tan (A + B)

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

Prove that
\[\frac{\tan A + \tan B}{\tan A - \tan B} = \frac{\sin \left( A + B \right)}{\sin \left( A - B \right)}\]

Prove that

Prove that

Prove that

Prove that:

Prove that:

Prove that:

Prove that \[\frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} = - 1\].

 If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].

If \[\tan A = \frac{m}{m - 1}\text{ and }\tan B = \frac{1}{2m - 1}\], then prove that \[A - B = \frac{\pi}{4}\].

Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]

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

Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]

Prove that:
\[\frac{\sin \left( A - B \right)}{\cos A \cos B} + \frac{\sin \left( B - C \right)}{\cos B \cos C} + \frac{\sin \left( C - A \right)}{\cos C \cos A} = 0\]

Prove that:

Prove that:
sin² B = sin² A + sin² (A − B) − 2 sin A cos B sin (A − B)

Prove that:
cos² A + cos² B − 2 cos A cos B cos (A + B) = sin² (A + B)

Prove that:
\[\frac{\tan \left( A + B \right)}{\cot \left( A - B \right)} = \frac{\tan^2 A - \tan^2 B}{1 - \tan^2 A \tan^2 B}\]

Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x

Prove that:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]

Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1

Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x

Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]

If tan A = x tan B, prove that
\[\frac{\sin \left( A - B \right)}{\sin \left( A + B \right)} = \frac{x - 1}{x + 1}\]

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

If x lies in the first quadrant and \[\cos x = \frac{8}{17}\], then prove that:

If tan x + \[\tan \left( x + \frac{\pi}{3} \right) + \tan \left( x + \frac{2\pi}{3} \right) = 3\], then prove that \[\frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} = 1\].

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

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

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

Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]

Prove that:

Prove that:

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

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

Find the maximum and minimum values of each of the following trigonometrical expression:

 12 sin x − 5 cos x 

Find the maximum and minimum values of each of the following trigonometrical expression: 

12 cos x + 5 sin x + 4 

Find the maximum and minimum values of each of the following trigonometrical expression: 

\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]

Find the maximum and minimum values of each of the following trigonometrical expression:

sin x − cos x + 1

Reduce each of the following expressions to the sine and cosine of a single expression: 

\[\sqrt{3} \sin x - \cos x\] 

Reduce each of the following expressions to the sine and cosine of a single expression: 

cos x − sin x 

Reduce each of the following expressions to the sine and cosine of a single expression: 

24 cos x + 7 sin x 

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

Prove that \[\left( 2\sqrt{3} + 3 \right) \sin x + 2\sqrt{3} \cos x\]  lies between \[- \left( 2\sqrt{3} + \sqrt{15} \right) \text{ and } \left( 2\sqrt{3} + \sqrt{15} \right)\]

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. 

If x cos θ = y cos \[\left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)\]then write the value of \[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\] 

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

Write the maximum value of 12 sin x − 9 sin² x. 

If 12 sin x − 9sin² x attains its maximum value at x = α, then write the value of sin α.

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

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

If \[\frac{\cos \left( x - y \right)}{\cos \left( x + y \right)} = \frac{m}{n}\]  then write the value of tan x tan y. 

If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.  

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

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

If tan \[\alpha = \frac{1}{1 + 2^{- x}}\] and \[\tan \beta = \frac{1}{1 + 2^{x + 1}}\] then write the value of α + β lying in the interval \[\left( 0, \frac{\pi}{2} \right)\] 

The value of \[\sin^2 \frac{5\pi}{12} - \sin^2 \frac{\pi}{12}\] 

If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to

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

If \[\tan A = \frac{a}{a + 1}\text{ and } \tan B = \frac{1}{2a + 1}\] 

If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =

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

tan 3A − tan 2A − tan A =

If A + B + C = π, then \[\frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C}\] is equal to

If \[\cos P = \frac{1}{7}\text{ and }\cos Q = \frac{13}{14}\], where P and Q both are acute angles. Then, the value of P − Q is

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

The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is

If tan θ₁ tan θ₂ = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]

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

If \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is 

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

If tan (π/4 + x) + tan (π/4 − x) = a, then tan² (π/4 + x) + tan² (π/4 − x) =

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

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

The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] is

If cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then

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

If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to