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 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(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
Prove the following:
sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove the following:
`(sin x - sin 3x)/(sin^2 x - cos^2 x) = 2sin x`
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:
cos (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 \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], 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:
tan (A + B)
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:
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
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 \[\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\] , 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 x
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
Prove that \[\left( 2\sqrt{3} + 3 \right) \sin x + 2\sqrt{3} \cos x\] lies between \[- \left( 2\sqrt{3} + \sqrt{15} \right) \text{ and } \left( 2\sqrt{3} + \sqrt{15} \right)\]
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 + C = π, then sec A (cos B cos C − sin B sin C) is equal to
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° 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 sin (π cos x) = cos (π sin x), then sin 2x = ______.
If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
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 tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`, then show that sinα + cosα = `sqrt(2)` cosθ.
[Hint: Express tanθ = `tan (alpha - pi/4) theta = alpha - pi/4`]
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]
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`]
The value of tan 75° - cot 75° is equal to ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
