Advertisements
Advertisements
प्रश्न
If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2
[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]
Advertisements
उत्तर
sin(θ + α) = a and sin(θ + β) = b
L.H.S = cos 2(α - β) - 4ab cos(α - β)
Using cos2x = 2cos2x - 1,
Let us solve,
⇒ LHS = 2cos2(α - β) - 1 - 4ab cos(α - β)
⇒ LHS = 2cos(α - β) {cos(α - β) - 2ab} - 1
Since,
cos(α - β) = cos{(θ + α) - (θ + β)}
cos(A - B) = cosA cosB + sinA sinB
⇒ cos(α - β) = cos(θ + α) cos(θ + β) + sin(θ + α) sin(θ + β)
Since, sin(θ + α) = a
⇒ cos(θ + α) = `sqrt(1 – sin^2(θ + alpha))`
= `sqrt(1 – "a"^2)`
Similarly,
cos(θ + β) = `sqrt(1 – b^2)`
Therefore,
cos(α - β) = `sqrt(1 - a^2) sqrt(1 - b^2) + ab`
Therefore,
L.H.S = `2{ab + sqrt(1 – a^2)(1 – b^2)}{ab + sqrt(1 – a^2)(1 – b^2) - 2ab} – 1`
⇒ L.H.S =`2{sqrt(1 – a^2)(1 – b^2) + ab}{sqrt(1 – a^2)(1 – b^2) – ab} - 1`
Using (x + y)(x - y) = x2 - y2
⇒ L.H.S = 2{(1 - a2)(1 - b2) - a2b2} - 1
⇒ L.H.S = 2{1 - a2 - b2 + a2b2} - 1
⇒ L.H.S = 2 - 2a2 - 2b2 - 1
⇒ L.H.S = 1 - 2a2 - 2b2 = RHS
Therefore,
We get,
cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2.
APPEARS IN
संबंधित प्रश्न
Prove that `2 sin^2 pi/6 + cosec^2 (7pi)/6 cos^2 pi/3 = 3/2`
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:
`tan 4x = (4tan x(1 - tan^2 x))/(1 - 6tan^2 x + tan^4 x)`
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 \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A + B)
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A + B)
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
Prove that
\[\frac{\tan A + \tan B}{\tan A - \tan B} = \frac{\sin \left( A + B \right)}{\sin \left( A - B \right)}\]
Prove that
Prove that \[\frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} = - 1\].
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]
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.
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\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
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
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
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 A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
If \[\cos P = \frac{1}{7}\text{ and }\cos Q = \frac{13}{14}\], where P and Q both are acute angles. Then, the value of P − Q is
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 cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
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 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
The maximum distance of a point on the graph of the function y = `sqrt(3)` sinx + cosx from x-axis is ______.
