हिंदी

Prove That: Cos2 a + Cos2 B − 2 Cos a Cos B Cos (A + B) = Sin2 (A + B)

Advertisements
Advertisements

प्रश्न

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

संक्षेप में उत्तर
Advertisements

उत्तर

\[\text{ LHS }= \cos^2 A + \cos^2 B - 2\cos A \cos B \cos\left( A + B \right)\]
\[ = \cos^2 A + 1 - \sin^2 B - 2\cos A \cos B \cos\left( A + B \right)\]
\[ = 1 + \cos^2 A - \sin^2 B - 2\cos A \cos B \cos\left( A + B \right)\]
\[ = 1 + \cos^2 A - \sin^2 B - 2\cos A \cos B \cos\left( A + B \right)\]
\[ = 1 + \cos\left( A + B \right)\cos\left( A - B \right) - 2\cos A \cos B \cos\left( A + B \right)\]
\[ = 1 + \cos\left( A + B \right)\left\{ \cos\left( A - B \right) - 2\cos A \cos B \right\}\]
\[ = 1 + \cos\left( A + B \right)\left( \cos A \cos B + \sin A \sin B - 2\cos A \cos B \right)\]
\[ = 1 + \cos\left( A + B \right)\left( - \cos A \cos B + \sin A \sin B \right)\]
\[ = 1 - \cos\left( A + B \right)\left( \cos A \cos B - \sin A \sin B \right)\]
\[ = 1 - \cos\left( A + B \right)\cos\left( A + B \right)\]
\[ = 1 - \cos^2 \left( A + B \right)\]
\[ = \sin^2 \left( A + B \right)\]
 = RHS
Hence proved.

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 7: Values of Trigonometric function at sum or difference of angles - Exercise 7.1 [पृष्ठ २०]

APPEARS IN

आर.डी. शर्मा Mathematics [English] Class 11
अध्याय 7 Values of Trigonometric function at sum or difference of angles
Exercise 7.1 | Q 16.5 | पृष्ठ २०

वीडियो ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्न

Prove the following: `cos (pi/4 xx x) cos (pi/4 - y) - sin (pi/4 -  x)sin (pi/4  - y) =  sin (x + y)`


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:

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`


Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x


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


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


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


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


Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (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 36° + tan 9° + tan 36° tan 9° = 1


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:

\[\cos \left( \frac{\pi}{6} + x \right) + \cos \left( \frac{\pi}{4} - x \right) + \cos \left( \frac{2\pi}{3} - x \right) = \left( \frac{\sqrt{3} - 1}{2} + \frac{1}{\sqrt{2}} \right)\frac{23}{17}\]

 


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


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


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

 12 sin x − 5 cos 


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 


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.  


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


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


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


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 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.


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


If tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`, then show that sinα + cosα = `sqrt(2)` cosθ.

[Hint: Express tanθ = `tan (alpha - pi/4) theta = alpha - pi/4`]


Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = `1/sqrt(2)`.


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

[Hint: A.M ≥ G.M.]


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


If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.


If tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×