Advertisements
Advertisements
प्रश्न
Prove that:
\[\sec\left( \frac{3\pi}{2} - x \right)\sec\left( x - \frac{5\pi}{2} \right) + \tan\left( \frac{5\pi}{2} + x \right)\tan\left( x - \frac{3\pi}{2} \right) = - 1 .\]
Advertisements
उत्तर
LHS = \[\sec\left( \frac{3\pi}{2} - x \right)\sec\left( x - \frac{5\pi}{2} \right) + \tan\left( \frac{5\pi}{2} + x \right)\tan\left( x - \frac{3\pi}{2} \right)\]
\[ = \sec\left( \frac{3\pi}{2} - x \right)\sec\left[ - \left( \frac{5\pi}{2} - x \right) \right] + \tan\left( \frac{5\pi}{2} + x \right)\tan\left[ - \left( \frac{3\pi}{2} - x \right) \right]\]
\[ = \sec\left( \frac{3\pi}{2} - x \right)\sec\left( \frac{5\pi}{2} - x \right) + \tan\left( \frac{5\pi}{2} + x \right)\left[ - \tan\left( \frac{3\pi}{2} - x \right) \right]\]
\[ = \sec\left( \frac{3\pi}{2} - x \right)\sec\left( \frac{5\pi}{2} - x \right) - \tan\left( \frac{5\pi}{2} + x \right)\tan\left( \frac{3\pi}{2} - x \right)\]
\[ = \sec\left( \frac{\pi}{2} \times 3 - x \right)\sec\left( \frac{\pi}{2} \times 5 - x \right) - \tan\left( \frac{\pi}{2} \times 5 + x \right)\tan\left( \frac{\pi}{2} \times 3 - x \right)\]
\[ = \left[ - cosec x \right]\left[ cosec x \right] - \left[ - \cot x \right]\cot x \]
\[ = - {cosec}^2 x + \cot^2 x\]
\[ = - \left[ {cosec}^2 x - \cot^2 x \right]\]
\[ = - 1\]
= RHS
Hence proved.
APPEARS IN
संबंधित प्रश्न
If \[x = \frac{2 \sin x}{1 + \cos x + \sin x}\], then prove that
If \[\cot x \left( 1 + \sin x \right) = 4 m \text{ and }\cot x \left( 1 - \sin x \right) = 4 n,\] \[\left( m^2 + n^2 \right)^2 = mn\]
If \[a = \sec x - \tan x \text{ and }b = cosec x + \cot x\], then shown that \[ab + a - b + 1 = 0\]
Prove that: \[\tan\frac{11\pi}{3} - 2\sin\frac{4\pi}{6} - \frac{3}{4} {cosec}^2 \frac{\pi}{4} + 4 \cos^2 \frac{17\pi}{6} = \frac{3 - 4\sqrt{3}}{2}\]
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
Prove that:
\[\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}\]
If \[0 < x < \frac{\pi}{2}\], and if \[\frac{y + 1}{1 - y} = \sqrt{\frac{1 + \sin x}{1 - \sin x}}\], then y is equal to
If \[\frac{\pi}{2} < x < \pi, \text{ then }\sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}}\] is equal to
Which of the following is incorrect?
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
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:
\[\sin x + \cos x = \sqrt{2}\]
Solve the following equation:
Solve the following equation:
`cosec x = 1 + cot x`
Solve the following equation:
\[2 \sin^2 x = 3\cos x, 0 \leq x \leq 2\pi\]
Solve the following equation:
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
Write the number of solutions of the equation tan x + sec x = 2 cos x in the interval [0, 2π].
Write the number of points of intersection of the curves
If \[3\tan\left( x - 15^\circ \right) = \tan\left( x + 15^\circ \right)\] \[0 < x < 90^\circ\], find θ.
If \[4 \sin^2 x = 1\], then the values of x are
If \[\cot x - \tan x = \sec x\], then, x is equal to
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°
cos 2x = 1 − 3 sin x
Solve the following equations:
cot θ + cosec θ = `sqrt(3)`
Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`
Choose the correct alternative:
If tan 40° = λ, then `(tan 140^circ - tan 130^circ)/(1 + tan 140^circ * tan 130^circ)` =
If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation ______.
Number of solutions of the equation tan x + sec x = 2 cosx lying in the interval [0, 2π] is ______.
In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.
