Advertisements
Advertisements
प्रश्न
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
Advertisements
उत्तर
Given:
y sin ϕ = x sin (2θ + ϕ)
\[\Rightarrow \frac{y}{x} = \frac{\sin\left( 2\theta + \phi \right)}{\sin\phi}\]
Applying componendo and dividendo:
\[ \Rightarrow \frac{y - x}{y + x} = \frac{\sin\left( 2\theta + \phi \right) - \sin\phi}{\sin\left( 2\theta + \phi \right) + \sin\phi}\]
\[ \Rightarrow \frac{y - x}{y + x} = \frac{2\sin\left( \frac{2\theta + \phi - \phi}{2} \right)\cos\left( \frac{2\theta + \phi + \phi}{2} \right)}{2\sin\left( \frac{2\theta + \phi + \phi}{2} \right)\cos\left( \frac{2\theta + \phi - \phi}{2} \right)}\]
\[ \Rightarrow \frac{y - x}{y + x} = \frac{2\sin \theta \cos\left( \theta + \phi \right)}{2\sin\left( \theta + \phi \right) \cos \theta}\]
\[ \Rightarrow \frac{y - x}{y + x} = \frac{\sin \theta \cos\left( \theta + \phi \right)}{\sin\left( \theta + \phi \right) \cos \theta}\]
\[ \Rightarrow \frac{y - x}{y + x} = \frac{\cot \left( \theta + \phi \right)}{\cot \theta}\]
\[ \Rightarrow \left( y - x \right) cot\theta = \left( y + x \right) cot\left( \theta + \phi \right)\]
APPEARS IN
संबंधित प्रश्न
Prove that:
Prove that:
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
Prove that:
Prove that:
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
Prove that:
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
cos 40° + cos 80° + cos 160° + cos 240° =
If sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
The value of cos 52° + cos 68° + cos 172° is
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in
Express the following as the sum or difference of sine or cosine:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
sin A + sin 2A
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
