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 the following:
sin2 6x – sin2 4x = sin 2x sin 10x
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 \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], 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 \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (A + B).
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], 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°
Prove that:
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
Prove that:
sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
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
Prove that:
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
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:
sin x − cos x + 1
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
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 A + B = C, then write the value of tan A tan B tan C.
If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
tan 3A − tan 2A − tan A =
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] 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 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 sinθ + cosθ = 1, then find the general value of θ.
Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = `1/sqrt(2)`.
If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = `(1 - m)/(1 + m) cot phi`
[Hint: Express `(cos(theta + Φ))/(cos(theta - Φ)) = m/1` and apply Componendo and Dividendo]
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
If tanα = `m/(m + 1)`, tanβ = `1/(2m + 1)`, then α + β is equal to ______.
The value of tan3A - tan2A - tanA is equal to ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
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`
