Advertisements
Advertisements
Question
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
Advertisements
Solution
\[\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha \sin\beta\]
\[ = \frac{2\cos\alpha\cos\beta - 2\sin\alpha \sin\beta + 2 - 2}{2}\]
\[ = \frac{2\cos\alpha\cos\beta - 2\sin\alpha \sin\beta + 2 - 2}{2}\]
\[=\frac{(sin\alpha - sin\beta )^2 + (\cos\alpha + \cos\beta )^2 - 2}{2}\]
\[=\frac{a^2 + b^2 - 2}{2}\]
APPEARS IN
RELATED QUESTIONS
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
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:
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:
sin 36° cos 9° + cos 36° sin 9°
Prove that
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Prove that:
\[\frac{\tan \left( A + B \right)}{\cot \left( A - B \right)} = \frac{\tan^2 A - \tan^2 B}{1 - \tan^2 A \tan^2 B}\]
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
If sin (α + β) = 1 and sin (α − β) \[= \frac{1}{2}\], where 0 ≤ α, \[\beta \leq \frac{\pi}{2}\], then find the values of tan (α + 2β) and tan (2α + β).
If sin α + sin β = a and cos α + cos β = b, show that
Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]
Prove that:
Prove that:
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
If tan (A + B) = p and tan (A − B) = q, then write the value of tan 2B.
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
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
If tan (A − B) = 1 and sec (A + B) = \[\frac{2}{\sqrt{3}}\], the smallest positive value of B is
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
Match each item given under column C1 to its correct answer given under column C2.
| C1 | C2 |
| (a) `(1 - cosx)/sinx` | (i) `cot^2 x/2` |
| (b) `(1 + cosx)/(1 - cosx)` | (ii) `cot x/2` |
| (c) `(1 + cosx)/sinx` | (iii) `|cos x + sin x|` |
| (d) `sqrt(1 + sin 2x)` | (iv) `tan x/2` |
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 sinθ + cosθ = 1, then find the general value of θ.
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`]
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
If f(x) = cos2x + sec2x, then ______.
[Hint: A.M ≥ G.M.]
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 sin(45° + θ) - cos(45° - θ) is ______.
