Advertisements
Advertisements
प्रश्न
If \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
उत्तर
Given:
\[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\]
Applying componendo and dividendo, we get
\[\frac{m + n}{m - n} = \frac{\sin\left( \theta + 2\alpha \right) + \sin\theta}{\sin\left( \theta + 2\alpha \right) - \sin\theta}\]
\[ \Rightarrow \frac{m + n}{m - n} = \frac{2\sin\left( \frac{\theta + 2\alpha + \theta}{2} \right)\cos\left( \frac{\theta + 2\alpha - \theta}{2} \right)}{2\sin\left( \frac{\theta + 2\alpha - \theta}{2} \right)\cos\left( \frac{\theta + 2\alpha + \theta}{2} \right)}\]
\[ \Rightarrow \frac{m + n}{m - n} = \frac{\sin\left( \theta + \alpha \right) \cos\alpha}{\sin\alpha \cos\left( \theta + \alpha \right)}\]
\[ \Rightarrow \frac{m + n}{m - n} = \tan\left( \theta + \alpha \right) \cot\alpha\]
APPEARS IN
संबंधित प्रश्न
Prove that:
Show that :
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Express each of the following as the product of sines and cosines:
cos 12x + cos 8x
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
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:
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 A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
cos 40° + cos 80° + cos 160° + cos 240° =
sin 163° cos 347° + sin 73° sin 167° =
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
Express the following as the product of sine and cosine.
cos 2A + cos 4A
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Evaluate:
sin 50° – sin 70° + sin 10°
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.
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)]`
