Advertisements
Advertisements
प्रश्न
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
विकल्प
tan B
cot B
tan 2 B
None of these
उत्तर
cot B
Since A,B and C are in A.P,
B - A = C - B
or, 2B = A + C
\[\frac{\sin A - \sin C}{\cos C - \cos A}\]
\[ = \frac{2\sin\left( \frac{A - C}{2} \right)\cos\left( \frac{A + C}{2} \right)}{- 2\sin\left( \frac{C + A}{2} \right)\sin\left( \frac{C - A}{2} \right)} \left[ \because \sin A - \sin B = 2\sin\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right) \text{ and }\cos A - \cos B = - 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = \frac{\sin\left( \frac{A - C}{2} \right)\cos\left( \frac{A + C}{2} \right)}{- \sin\left( \frac{A + C}{2} \right)\sin\left( \frac{C - A}{2} \right)}\]
\[= \frac{\sin\left( \frac{A - C}{2} \right)\cos\left( \frac{A + C}{2} \right)}{\sin\left( \frac{A + C}{2} \right)\sin\left( \frac{A - C}{2} \right)}\]
\[ = \frac{\cos\left( \frac{A + C}{2} \right)}{\sin\left( \frac{A + C}{2} \right)}\]
\[ = \frac{\cos B}{\sin B}\]
\[ = \cot B\]
APPEARS IN
संबंधित प्रश्न
Prove that:
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 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
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
Prove that:
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Prove that:
Prove that:
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:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
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 \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
If A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
The value of cos 52° + cos 68° + cos 172° is
The value of sin 78° − sin 66° − sin 42° + sin 60° is ______.
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If \[\tan\alpha = \frac{x}{x + 1}\] and
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
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
