English

Prove the following: sinx- sinycosx+cosy=tan x-y2

Advertisements
Advertisements

Question

Prove the following:

`(sin x -  siny)/(cos x + cos y)= tan  (x -y)/2`

Sum
Advertisements

Solution

We have, बायाँ पक्ष = `(sin x -  siny)/(cos x + cos y)`

= `(2sin ((x - y )/2) cos ((x + y)/2))/(2cos ((x - y)/2) cos ((x + y)/2)`

= `(sin (x -y)/2)/(cos (x - y)/2)` 

= tan `(x - y)/2` = दायाँ पक्ष।

shaalaa.com
  Is there an error in this question or solution?
Chapter 3: Trigonometric Functions - EXERCISE 3.3 [Page 67]

APPEARS IN

NCERT Mathematics [English] Class 11
Chapter 3 Trigonometric Functions
EXERCISE 3.3 | Q 18. | Page 67

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Prove that: `sin^2  pi/6 + cos^2  pi/3 - tan^2  pi/4 = -1/2`


Prove that  `cot^2  pi/6 + cosec  (5pi)/6 + 3 tan^2  pi/6 = 6`


Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`


Prove the following:

`(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) =  cot^2 x`


Prove the following:

cos2 2x – cos2 6x = sin 4x sin 8x


Prove the following:

`(sin x - sin 3x)/(sin^2 x - cos^2 x) =  2sin x`


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


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


Prove that:

\[\sin\left( \frac{4\pi}{9} + 7 \right)\cos\left( \frac{\pi}{9} + 7 \right) - \cos\left( \frac{4\pi}{9} + 7 \right)\sin\left( \frac{\pi}{9} + 7 \right) = \frac{\sqrt{3}}{2}\]

 


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


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


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


Prove that:

\[\frac{1}{\cos \left( x - a \right) \cos \left( a - b \right)} = \frac{\tan \left( x - b \right) - \tan \left( x - a \right)}{\sin \left( a - b \right)}\]

 


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


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 


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

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


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 \[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\] 


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


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 \[\left( 0, \frac{\pi}{2} \right)\] 


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


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


\[\frac{\cos 10^\circ + \sin 10^\circ}{\cos 10^\circ - \sin 10^\circ} =\]

 


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

 

If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = `(2ac)/(a^2 - c^2)`.


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 `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].


Find the general solution of the equation `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2

[Hint: Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα which gives tanα = `tan(pi/4 - pi/6)` α = `pi/12`]


If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.


The value of sin(45° + θ) - cos(45° - θ) is ______.


If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.


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


If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×