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:
Show that :
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
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 12x + sin 4x
Express each of the following as the product of sines and cosines:
sin 5x − sin x
Express each of the following as the product of sines and cosines:
cos 12x + cos 8x
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Prove that:
Prove that:
Prove that:
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
If (cos α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ.
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
cos 40° + cos 80° + cos 160° + cos 240° =
The value of cos 52° + cos 68° + cos 172° is
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
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
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
