Advertisements
Advertisements
प्रश्न
If \[T_n = \sin^n x + \cos^n x\], prove that \[\frac{T_3 - T_5}{T_1} = \frac{T_5 - T_7}{T_3}\]
Advertisements
उत्तर
LHS:\[\frac{T_3 - T_5}{T_1} = \frac{\left( \sin^3 x + \cos^3 x \right) - \left( \sin^5 x + \cos^5 x \right)}{\sin x + \cos x}\]
\[ = \frac{\sin^3 x - \sin^5 x + \cos^3 x - \cos^5 x}{\sin x + \cos x}\]
\[ = \frac{\sin^3 x\left( 1 - \sin^2 x \right) + \cos^3 x\left( 1 - \cos^2 x \right)}{\sin x + \cos x}\]
\[ = \frac{\sin^3 x . \cos^2 x + c {os}^3 x . \sin^2 x}{\sin x + \cos x}\]
\[ = \frac{\sin^2 x . \cos^2 x\left( \sin x + cos x \right)}{\sin x + \cos x}\]
\[ = \sin^2 x . \cos^2 x\]
RHS: \[\frac{T_5 - T_7}{T_3}\]
\[ = \frac{\left( \sin^5 x + \cos^5 x \right) - \left( \sin^7 x + \cos^7 x \right)}{\sin^3 x + \cos^3 x}\]
\[ = \frac{\sin^5 x - si n^7 x + \cos^5 x - \cos^7 x}{\sin^3 x + \cos^3 x}\]
\[ = \frac{\sin^5 x\left( 1 - \sin^2 x \right) + \cos^5 x\left( 1 - \cos^2 x \right)}{\sin^3 x + \cos^3 x}\]
\[ = \frac{\sin^5 x \cos^2 x + \cos^5 x \sin^2 x}{\sin^3 x + \cos^3 x}\]
\[ = \sin^2 x . \cos^2 x\]
LHS = RHS
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the principal and general solutions of the equation sec x = 2
Find the general solution of the equation cos 4 x = cos 2 x
Find the general solution of the equation sin 2x + cos x = 0
If \[\tan x = \frac{b}{a}\] , then find the values of \[\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}\].
Prove that:
\[\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}\]
If tan \[x = - \frac{1}{\sqrt{5}}\] and θ lies in the IV quadrant, then the value of cos x is
If \[\frac{3\pi}{4} < \alpha < \pi, \text{ then }\sqrt{2\cot \alpha + \frac{1}{\sin^2 \alpha}}\] is equal to
If tan A + cot A = 4, then tan4 A + cot4 A is equal to
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
The value of \[\cos1^\circ \cos2^\circ \cos3^\circ . . . \cos179^\circ\] is
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:
Solve the following equation:
Solve the following equation:
\[\sec x\cos5x + 1 = 0, 0 < x < \frac{\pi}{2}\]
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 values of x in [0, 2π] that satisfy the equation \[\sin x - \cos x = \frac{1}{4}\].
A solution of the equation \[\cos^2 x + \sin x + 1 = 0\], lies in the interval
The general value of x satisfying the equation
\[\sqrt{3} \sin x + \cos x = \sqrt{3}\]
The smallest positive angle which satisfies the equation
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
The equation \[3 \cos x + 4 \sin x = 6\] has .... solution.
The solution of the equation \[\cos^2 x + \sin x + 1 = 0\] lies in the interval
If \[\cos x = - \frac{1}{2}\] and 0 < x < 2\pi, then the solutions are
Find the principal solution and general solution of the following:
cot θ = `sqrt(3)`
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:
sin θ + cos θ = `sqrt(2)`
Solve the equation sin θ + sin 3θ + sin 5θ = 0
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 a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a2 + b2 = m2 + n2
