Advertisements
Advertisements
प्रश्न
Prove that:
cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
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.
APPEARS IN
संबंधित प्रश्न
Prove that `cot^2 pi/6 + cosec (5pi)/6 + 3 tan^2 pi/6 = 6`
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
`(cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x`
Prove the following:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
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 \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (A + B).
Prove that
Prove that
Prove that:
sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin 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:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) \[\frac{1}{a} - \frac{1}{b}\].
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 α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
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:
\[\sqrt{3} \sin x - \cos x\]
Show that sin 100° − sin 10° is positive.
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
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 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 \[\left( 0, \frac{\pi}{2} \right)\]
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 tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
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 69° + tan 66° − tan 69° tan 66° = 2k, then k =
Express the following as the sum or difference of sines and cosines:
2 sin 4x sin 3x
If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
If sinθ + cosθ = 1, then find the general value of θ.
If tanα = `m/(m + 1)`, tanβ = `1/(2m + 1)`, then α + β is equal to ______.
The value of sin(45° + θ) - cos(45° - θ) is ______.
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 tanθ + tan2θ + `sqrt(3)` tanθ tan2θ = `sqrt(3)`, then θ = `("n"pi)/3 + pi/9`
