Advertisements
Advertisements
Question
Prove that:
Advertisements
Solution
Consider LHS:
\[ \frac{\sin A + \sin B}{\sin A - \sin B}\]
\[ = \frac{2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)}{2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)} \left\{ \because \sin A + \sin B = 2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right), and \sin A - \sin B = 2\sin \left( \frac{A - B}{2} \right) \cos\left( \frac{A + B}{2} \right) \right\}\]
\[ = \frac{\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)}{\sin \left( \frac{A - B}{2} \right) \cos \left( \frac{A + B}{2} \right)}\]
\[ = \tan \left( \frac{A + B}{2} \right) cot \left( \frac{A - B}{2} \right)\]
= RHS
Hence, LHS = RHS.
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Prove that
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
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 (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 \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
If \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
If (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 \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
The value of sin 50° − sin 70° + sin 10° is equal to
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 (7"A")/3 sin (5"A")/3`
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.
cos 2A + cos 4A
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:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Find the value of tan22°30′. `["Hint:" "Let" θ = 45°, "use" tan theta/2 = (sin theta/2)/(cos theta/2) = (2sin theta/2 cos theta/2)/(2cos^2 theta/2) = sintheta/(1 + costheta)]`
