Advertisements
Advertisements
प्रश्न
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Advertisements
उत्तर
L.H.S = `tanx tan(pi/3 - x) tan (pi/3 + x)`
= `tanx . (sin(pi/3 - x))/(cos(pi/3 - x)).sin(pi/3 + x)/(cos(pi/3 + x))`
= `(sinx . sin(pi/3 - x). sin(pi/3 + x))/(cosx . cos(pi/3 - x) . cos(pi/3 + x))`
= `(sinx . (sin^2 pi/3 - sin^2x))/(cosx . (cos^2 pi/3 - sin^2x))`
= `sinx/cosx((sqrt3/2)^2 - sin^2x)/((1/2)^2 - sin^2x)`
= `sinx/cosx ((3/4) - sin^2x)/((1/4) - sin^2x)`
= `sinx/cosx ((3 - 4sin^2x)/(1 - 4sin^2x))`
= `sinx/cosx ((3 - 4sin^2x)/(1 - 4(1 - cos^2x)))`
= `sinx/cosx ((3 - 4sin^2x)/(4 cos^2x - 3))`
= `(3 sinx - 4 sin^3x)/(4cos^2 - 3cosx)`
= `(sin3x)/(cos3x)`
= `tanx`
APPEARS IN
संबंधित प्रश्न
Show that :
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 5x − sin x
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
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:
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 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 \[xy + yz + zx = 0\]
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
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.
sin 6θ – sin 2θ
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Evaluate-
cos 20° + cos 100° + cos 140°
Evaluate:
sin 50° – sin 70° + sin 10°
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
