Advertisements
Advertisements
प्रश्न
Prove that:
\[\sin^2 \frac{\pi}{18} + \sin^2 \frac{\pi}{9} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{4\pi}{9} = 2\]
Advertisements
उत्तर
LHS = \[\sin^2 \frac{\pi}{18} + \sin^2 \frac{\pi}{9} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{4\pi}{9}\]
\[ = \sin^2 \frac{\pi}{18} + \sin^2 \frac{2\pi}{18} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{8\pi}{18}\]
\[ = \sin^2 \frac{\pi}{18} + \sin^2 \frac{2\pi}{18} + \sin^2 \left( \frac{7\pi}{18} \right) + \sin^2 \left( \frac{8\pi}{18} \right)\]
\[ = \sin^2 \frac{\pi}{18} + \sin^2 \frac{2\pi}{18} + \sin^2 \left( \frac{\pi}{2} - \frac{2\pi}{18} \right) + \sin^2 \left( \frac{\pi}{2} - \frac{\pi}{18} \right)\]
\[ = \sin^2 \frac{\pi}{18} + \sin^2 \frac{2\pi}{18} + \cos^2 \frac{2\pi}{18} + \cos^2 \frac{\pi}{18}\]
\[ = \sin^2 \frac{\pi}{18} + \cos^2 \frac{\pi}{18} + \sin^2 \frac{2\pi}{18} + \cos^2 \frac{2\pi}{18}\]
\[ = 1 + 1\]
\[ = 2\]
= RHS
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the general solution of the equation sin 2x + cos x = 0
Prove that
Prove that
In a ∆A, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that cos(180° − A) + cos (180° + B) + cos (180° + C) − sin (90° + D) = 0
Find x from the following equations:
\[x \cot\left( \frac{\pi}{2} + \theta \right) + \tan\left( \frac{\pi}{2} + \theta \right)\sin \theta + cosec\left( \frac{\pi}{2} + \theta \right) = 0\]
If \[\frac{\pi}{2} < x < \frac{3\pi}{2},\text{ then }\sqrt{\frac{1 - \sin x}{1 + \sin x}}\] 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 A lies in second quadrant 3tan A + 4 = 0, then the value of 2cot A − 5cosA + sin A is equal to
If \[f\left( x \right) = \cos^2 x + \sec^2 x\], then
Which of the following is correct?
Find the general solution of the following equation:
Find the general solution of the following equation:
Solve the following equation:
\[\sin^2 x - \cos x = \frac{1}{4}\]
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
4sinx cosx + 2 sin x + 2 cosx + 1 = 0
Solve the following equation:
3tanx + cot x = 5 cosec x
Solve the following equation:
3 – 2 cos x – 4 sin x – cos 2x + sin 2x = 0
Write the general solutions of tan2 2x = 1.
Write the number of points of intersection of the curves
If a is any real number, the number of roots of \[\cot x - \tan x = a\] in the first quadrant is (are).
A solution of the equation \[\cos^2 x + \sin x + 1 = 0\], lies in the interval
The number of solution in [0, π/2] of the equation \[\cos 3x \tan 5x = \sin 7x\] is
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
If \[\cos x = - \frac{1}{2}\] and 0 < x < 2\pi, then the solutions are
The number of values of x in the interval [0, 5 π] satisfying the equation \[3 \sin^2 x - 7 \sin x + 2 = 0\] is
Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`
Find the principal solution and general solution of the following:
tan θ = `- 1/sqrt(3)`
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
2 cos2x + 1 = – 3 cos x
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
cos 2x = 1 − 3 sin x
Solve the following equations:
sin 2θ – cos 2θ – sin θ + cos θ = θ
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:
If f(θ) = |sin θ| + |cos θ| , θ ∈ R, then f(θ) is in the interval
The minimum value of 3cosx + 4sinx + 8 is ______.
