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
Find the principal and general solutions of the equation `tan x = sqrt3`
Find the general solution of cosec x = –2
Find the general solution for each of the following equations sec2 2x = 1– tan 2x
Find the general solution of the equation sin x + sin 3x + sin 5x = 0
If \[a = \sec x - \tan x \text{ and }b = cosec x + \cot x\], then shown that \[ab + a - b + 1 = 0\]
Prove that:
\[\frac{\cos (2\pi + x) cosec (2\pi + x) \tan (\pi/2 + x)}{\sec(\pi/2 + x)\cos x \cot(\pi + x)} = 1\]
In a ∆ABC, prove that:
Prove that:
sin6 A + cos6 A + 3 sin2 A cos2 A =
If x sin 45° cos2 60° = \[\frac{\tan^2 60^\circ cosec30^\circ}{\sec45^\circ \cot^{2^\circ} 30^\circ}\], then x =
If \[f\left( x \right) = \cos^2 x + \sec^2 x\], then
Find the general solution of the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
`cosec x = 1 + cot x`
Solve the following equation:
\[\sec x\cos5x + 1 = 0, 0 < x < \frac{\pi}{2}\]
Solve the following equation:
3tanx + cot x = 5 cosec x
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 \[2 \sin^2 x = 3\cos x\]. where \[0 \leq x \leq 2\pi\], then find the value of x.
If \[\tan px - \tan qx = 0\], then the values of θ form a series in
The general solution of the equation \[7 \cos^2 x + 3 \sin^2 x = 4\] is
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
If \[e^{\sin x} - e^{- \sin x} - 4 = 0\], then x =
Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
2 sin2x + 1 = 3 sin x
Solve the following equations:
sin 2θ – cos 2θ – sin θ + cos θ = θ
Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`
Solve the following equations:
cos 2θ = `(sqrt(5) + 1)/4`
Solve the following equations:
2cos 2x – 7 cos x + 3 = 0
Choose the correct alternative:
If tan 40° = λ, then `(tan 140^circ - tan 130^circ)/(1 + tan 140^circ * tan 130^circ)` =
Choose the correct alternative:
If cos pθ + cos qθ = 0 and if p ≠ q, then θ is equal to (n is any integer)
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 sin α + cos α = b, then sin 2α is equal to
If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation ______.
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.
Number of solutions of the equation tan x + sec x = 2 cosx lying in the interval [0, 2π] is ______.
