English

If Tan θ1 Tan θ2 = K, Then Cos ( θ 1 − θ 2 ) Cos ( θ 1 + θ 2 ) =

Advertisements
Advertisements

Question

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

Options

  • \[\frac{1 + k}{1 - k}\]

     

  • \[\frac{1 - k}{1 + k}\]

     

  • \[\frac{k + 1}{k - 1}\]

     

  • \[\frac{k - 1}{k + 1}\]

     

MCQ
Advertisements

Solution

\[\frac{1 + k}{1 - k}\]

\[\frac{\cos( \theta_1 - \theta_2 )}{\cos( \theta_1 + \theta_2 )}\]

\[ = \frac{\cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2}{\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2}\]

\[\text{ Dividing numerator and denominator by }\cos \theta_1 \cos \theta_2 ,\text{ we get }:\]
\[\frac{1 + \tan \theta_1 \tan \theta_2}{1 - \tan \theta_1 \tan \theta_2}\]
\[ = \frac{1 + k}{1 - k}\]

shaalaa.com
  Is there an error in this question or solution?
Chapter 7: Values of Trigonometric function at sum or difference of angles - Exercise 7.4 [Page 28]

APPEARS IN

R.D. Sharma Mathematics [English] Class 11
Chapter 7 Values of Trigonometric function at sum or difference of angles
Exercise 7.4 | Q 13 | Page 28

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Prove the following:

`cos ((3pi)/ 2 + x ) cos(2pi + x) [cot ((3pi)/2 - x) + cot (2pi + x)]= 1`


Prove the following:

sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x


Prove the following:

sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x


Prove that: `(cos x  + cos y)^2 + (sin x - sin y )^2 =  4 cos^2  (x + y)/2`


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)


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


Prove that:

\[\sin\left( \frac{\pi}{3} - x \right)\cos\left( \frac{\pi}{6} + x \right) + \cos\left( \frac{\pi}{3} - x \right)\sin\left( \frac{\pi}{6} + x \right) = 1\]

 


Prove that:

\[\sin\left( \frac{3\pi}{8} - 5 \right)\cos\left( \frac{\pi}{8} + 5 \right) + \cos\left( \frac{3\pi}{8} - 5 \right)\sin\left( \frac{\pi}{8} + 5 \right) = 1\]

 


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


Prove that:

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

 


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


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


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 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).

 

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

\[\sin \left( \alpha + \beta \right) = \frac{2ab}{a^2 + b^2}\]

 


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\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]

 

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


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


tan 3A − tan 2A − tan A =


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

 

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


Express the following as the sum or difference of sines and cosines:
2 sin 4x sin 3x


Express the following as the sum or difference of sines and cosines:
 2 cos 7x cos 3x


If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and Φ is their difference, then show that sin θ = `(k + 1)/(k - 1)` sin Φ


If f(x) = cos2x + sec2x, then ______.

[Hint: A.M ≥ G.M.]


The value of tan 75° - cot 75° is equal to ______.


If sinθ + cosθ = 1, then the value of sin2θ is equal to ______.


Given x > 0, the values of f(x) = `-3cos sqrt(3 + x + x^2)` lie in the interval ______.


State whether the statement is True or False? Also give justification.

If tan(π cosθ) = cot(π sinθ), then `cos(theta - pi/4) = +- 1/(2sqrt(2))`.


In the following match each item given under the column C1 to its correct answer given under the column C2:

Column A Column B
(a) sin(x + y) sin(x – y) (i) cos2x – sin2y
(b) cos (x + y) cos (x – y) (ii) `(1 - tan theta)/(1 + tan theta)`
(c) `cot(pi/4 + theta)` (iii) `(1 + tan theta)/(1 - tan theta)`
(d) `tan(pi/4 + theta)` (iv) sin2x – sin2y

Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×