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
$$\frac{\tan A+\tan B}{\tan A-\tan B}=\frac{\sin (A+B)}{\sin (A-B)}$$

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: $$\frac{\sin (A+B)+\sin (A-B)}{\cos (A+B)+\cos (A-B)}=\tan A$$

Prove that:
$$\frac{\sin (A-B)}{\cos A\cos B}+\frac{\sin (B-C)}{\cos B\cos C}+\frac{\sin (C-A)}{\cos C\cos A}=0$$

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:
$$\frac{\tan (A+B)}{\cot (A-B)}=\frac{{\tan }^{2}A-{\tan }^{2}B}{1-{\tan }^{2}A{\tan }^{2}B}$$

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
$$\frac{\sin (A-B)}{\sin (A+B)}=\frac{x-1}{x+1}$$

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 + $$\tan (x+\frac{\pi }{3})+\tan (x+\frac{2\pi }{3})=3$$, 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, $$\beta \le \frac{\pi }{2}$$, 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:
$$\frac{1}{\sin (x-a)\sin (x-b)}=\frac{\cot (x-a)-\cot (x-b)}{\sin (a-b)}$$

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 $$\varphi$$ is their difference, then show that$$\sin \theta =\frac{\lambda +1}{\lambda -1}\sin \varphi$$

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: 

$$5\cos x+3\sin (\frac{\pi }{6}-x)+4$$

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 $$(2\sqrt{3}+3)\sin x+2\sqrt{3}\cos x$$  lies between $$-(2\sqrt{3}+\sqrt{15})\text{ and }(2\sqrt{3}+\sqrt{15})$$

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 $$(\theta +\frac{2\pi }{3})=z\cos (\theta +\frac{4\pi }{3})$$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 $$(x+\frac{\pi }{3})+3$$ lies. 

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

If $$\frac{\cos (x-y)}{\cos (x+y)}=\frac{m}{n}$$  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 $$(0,\frac{\pi }{2})$$ 

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 $${\cos }^2(\frac{\pi }{6}+x)-{\sin }^2(\frac{\pi }{6}-x)$$ is

If tan θ₁ tan θ₂ = k, then $$\frac{\cos ({\theta }_1-{\theta }_2)}{\cos ({\theta }_1+{\theta }_2)}=$$

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

If $$\tan \theta =\frac{1}{2}$$ and $$\tan \varphi =\frac{1}{3}$$, then the value of $$\tan \varphi =\frac{1}{3}$$ 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 $${\sin }^2(\frac{2\pi }{3}+x)+{\sin }^2(\frac{2\pi }{3}-x)$$ 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