Advertisements
Advertisements
Question
Prove the following:
tan6° tan42° tan66° tan78° = 1
Solution
L.H.S. = tan6° tan42° tan66° tan78°
= `sin6^circ/cos6^circ * sin42^circ/cos42^circ *sin66^circ/cos66^circ *sin78^circ/cos78^circ`
= `((2sin66^circ sin6^circ)(2sin78^circ sin42^circ))/((2cos66^circ cos6^circ)(2cos78^circ cos42^circ)`
= `(cos(66^circ - 6^circ) - cos(66^circ + 6^circ))/(cos(66^circ + 6^circ) + cos(66^circ - 6^circ)) * (cos(78^circ - 42^circ) - cos(78^circ + 42^circ))/(cos(78^circ + 42^circ) + cos(78^circ - 42^circ))`
= `((cos60^circ - cos72^circ)(cos36^circ - cos120^circ))/((cos60^circ + cos72^circ)(cos36^circ + cos120^circ)`
= `((cos60^circ - sin18^circ)(cos36^circ + sin30^circ))/((cos60^circ + sin18^circ)(cos36^circ - sin30^circ)` ...[∵ cos(90° + θ) = – sin θ]
= `((1/2 - (sqrt(5) - 1)/4)((sqrt(5) + 1)/4 + 1/2))/((1/2 + (sqrt(5) - 1)/4)((sqrt(5) + 1)/4 - 1/2))`
= `(9 - 5)/(5 - 1)`
= 1
= R.H.S.
APPEARS IN
RELATED QUESTIONS
Find the values of:
cos 75°
Find the values of:
tan 105°
Find the value of :
sin (495°)
Find the value of :
cos 315°
Find the value of :
sec 240°
Find the value of :
sec (– 855°)
Prove the following:
sec 840° . cot (– 945°) + sin 600° tan (– 690°) = `3/2`
Prove the following:
`("cosec"(90^circ - x)sin(180^circ - x)cot(360^circ - x))/(sec(180^circ + x)tan(90^circ + x)sin(-x))` = 1
Prove the following:
`(sin^3(pi + x)sec^2(pi - x)tan(2pi - x))/(cos^2(pi/2 + x)sin(pi - x)"cosec"^2 - x)` = tan3x
Select the correct option from the given alternatives :
Let 0 < A, B < `pi/2` satisfying the equation 3 sin2A + 2 sin2B = 1 and 3 sin 2A − 2 sin 2B = 0 then A + 2B is equal to ______
Prove the following:
tan 20° tan 80° cot 50° = `sqrt(3)`
Prove the following:
sin 20° sin 40° sin 80° = `sqrt(3)/8`
Prove the following:
sin 18° = `(sqrt(5) - 1)/4`
Prove the following:
cos 36° = `(sqrt(5) + 1)/4`
Prove the following:
sin 36° = `(sqrt(10 - 2sqrt(5)))/4`
If a = sin 175°+ cos 175°, then ______.
If f(x) = `(2"x" + 3)/(3"x" - 2)`, `"x" ≠ 2/3`, then the function fof is ____________.
If θ = `(17π)/3` then, tan θ – cot θ = ______.
`(1 - 2[cos 60^circ - cos 80^circ])/(2 sin 10^circ)` = ______.
The value of sin 495° is ______.
The value of `cos((41π)/4)` is ______.
Find the value of `cos ((29 π)/3)`.
If `cosA/3 = cosB/4 = 1/5, - π/2 < A < 0` and `- π/2 < B < 0`, then the value of 2 sin A + 4 sin B is ______.
sin2 17.5° + sin2 72.5° is equal to ______.
If ΔABC is a right angled at C, then tan A + tan B is equal to ______.
In a ΔPQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then ∠R is equal to ______.
The value of sin 135° cosec 225° tan 150° cot 315° is ______.
The value of `(cos(90^circ + θ) sec(-θ)tan(180^circ - θ))/(sec(360^circ - θ)sin(180^circ + θ)cot(90^circ - θ))` is ______.
The value of sin 150° cos 120° + cos 330° sin 660° is ______.
The value of `2 sin^2 π/6 + "cosec"^2 (7π)/6 cos^2 π/3` is ______.
cos 1°. cos 2°. cos 3° ...... cos 179° = ______.
sin (270° – θ) sin (90° – θ) – cos ( 270° – θ) cos (90° + θ) is ______.
