Advertisements
Advertisements
प्रश्न
The value of \[\tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ\] is
विकल्प
0
1
- \[\frac{1}{2}\]
not defined
Advertisements
उत्तर
We know that,
\[\tan\left( 90^\circ - \theta \right) = \cot\theta\]
So,
\[\tan89^\circ = \tan\left( 90^\circ - 1^\circ \right) = \cot1^\circ\]
\[\tan88^\circ = \tan\left( 90^\circ - 2^\circ \right) = \cot2^\circ\]
\[\tan87^\circ = \tan\left( 90^\circ - 3^\circ \right) = \cot3^\circ\]
. . . .
. . . .
\[\tan46^\circ = \tan\left( 90^\circ - 44^\circ \right) = \cot44^\circ\]
\[\therefore \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ\]
\[ = \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan44^\circ \tan45^\circ \tan46^\circ . . . \tan87^\circ \tan88^\circ \tan89^\circ\]
\[ = \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan44^\circ \tan45^\circ \cot44^\circ. . . \cot3^\circ \cot2^\circ \cot1^\circ\]
\[ = \left( \tan1^\circ\cot1^\circ \right)\left( \tan2^\circ\cot2^\circ \right) \left( \tan3^\circ\cot3^\circ \right) . . . \left( \tan44^\circ\cot44^\circ \right)\tan45^\circ\]
\[ = 1 \left( \tan45^\circ = 1\text{ and }\tan\theta\cot\theta = 1 \right)\]
Hence, the correct answer is option 1.
APPEARS IN
संबंधित प्रश्न
Find the principal and general solutions of the equation `cot x = -sqrt3`
If \[\sin x = \frac{a^2 - b^2}{a^2 + b^2}\], then the values of tan x, sec x and cosec x
Prove that:
Prove that:
\[\sin^2 \frac{\pi}{18} + \sin^2 \frac{\pi}{9} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{4\pi}{9} = 2\]
Find x from the following equations:
\[cosec\left( \frac{\pi}{2} + \theta \right) + x \cos \theta \cot\left( \frac{\pi}{2} + \theta \right) = \sin\left( \frac{\pi}{2} + \theta \right)\]
Find x from the following equations:
\[x \cot\left( \frac{\pi}{2} + \theta \right) + \tan\left( \frac{\pi}{2} + \theta \right)\sin \theta + cosec\left( \frac{\pi}{2} + \theta \right) = 0\]
Prove that:
\[\sin\frac{13\pi}{3}\sin\frac{8\pi}{3} + \cos\frac{2\pi}{3}\sin\frac{5\pi}{6} = \frac{1}{2}\]
If sec \[x = x + \frac{1}{4x}\], then sec x + tan x =
If tan x + sec x = \[\sqrt{3}\], 0 < x < π, then x is equal to
If tan \[x = - \frac{1}{\sqrt{5}}\] and θ lies in the IV quadrant, then the value of cos x is
The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is
If tan θ + sec θ =ex, then cos θ equals
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
\[\sec x\cos5x + 1 = 0, 0 < x < \frac{\pi}{2}\]
Solve the following equation:
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
Solve the following equation:
3 – 2 cos x – 4 sin x – cos 2x + sin 2x = 0
Solve the following equation:
3sin2x – 5 sin x cos x + 8 cos2 x = 2
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
Write the number of solutions of the equation
\[4 \sin x - 3 \cos x = 7\]
If \[2 \sin^2 x = 3\cos x\]. where \[0 \leq x \leq 2\pi\], then find the value of x.
If \[\cos x + \sqrt{3} \sin x = 2,\text{ then }x =\]
The general solution of the equation \[7 \cos^2 x + 3 \sin^2 x = 4\] is
The smallest positive angle which satisfies the equation
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
The solution of the equation \[\cos^2 x + \sin x + 1 = 0\] lies in the interval
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`
Solve the following equations:
2cos 2x – 7 cos x + 3 = 0
Choose the correct alternative:
If tan α and tan β are the roots of x2 + ax + b = 0 then `(sin(alpha + beta))/(sin alpha sin beta)` is equal to
Choose the correct alternative:
`(cos 6x + 6 cos 4x + 15cos x + 10)/(cos 5x + 5cs 3x + 10 cos x)` is equal to
Solve 2 tan2x + sec2x = 2 for 0 ≤ x ≤ 2π.
Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
