Advertisements
Advertisements
Question
If tan x = \[x - \frac{1}{4x}\], then sec x − tan x is equal to
Options
\[- 2x, \frac{1}{2x}\]
\[- \frac{1}{2x}, 2x\]
2x
\[2x, \frac{1}{2x}\]
Advertisements
Solution
\[- 2x, \frac{1}{2x}\]
We have,
\[\tan x = x - \frac{1}{4x}\]
\[ \Rightarrow se c^2 x = 1 + \tan^2 x\]
\[ \Rightarrow se c^2 x = 1 + \left( x - \frac{1}{4x} \right)^2 \]
\[ \Rightarrow se c^2 x = x^2 + \frac{1}{16 x^2} + \frac{1}{2}\]
\[ \Rightarrow se c^2 x = \left( x + \frac{1}{4x} \right)^2 \]
\[ \therefore secx = \pm \left( x + \frac{1}{4x} \right)\]
\[ \Rightarrow secx - \tan x = \left( x + \frac{1}{4x} \right) - \left( x - \frac{1}{4x} \right) or - \left( x + \frac{1}{4x} \right) - \left( x - \frac{1}{4x} \right)\]
\[ = \frac{1}{2x}\text{ or }- 2x\]
APPEARS IN
RELATED QUESTIONS
If \[\tan x = \frac{b}{a}\] , then find the values of \[\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}\].
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 \[ab + a - b + 1 = 0\]
If \[T_n = \sin^n x + \cos^n x\], prove that \[2 T_6 - 3 T_4 + 1 = 0\]
If \[T_n = \sin^n x + \cos^n x\], prove that \[6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0\]
Prove that: tan 225° cot 405° + tan 765° cot 675° = 0
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:
\[cosec\left( \frac{\pi}{2} + \theta \right) + x \cos \theta \cot\left( \frac{\pi}{2} + \theta \right) = \sin\left( \frac{\pi}{2} + \theta \right)\]
If sec \[x = x + \frac{1}{4x}\], then sec x + tan x =
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
If tan \[x = - \frac{1}{\sqrt{5}}\] and θ lies in the IV quadrant, then the value of cos x is
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
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:
Solve the following equation:
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
Solve the following equation:
cosx + sin x = cos 2x + sin 2x
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
Write the number of solutions of the equation
\[4 \sin x - 3 \cos x = 7\]
Write the general solutions of tan2 2x = 1.
If cos x = k has exactly one solution in [0, 2π], then write the values(s) of k.
Write the number of values of x in [0, 2π] that satisfy the equation \[\sin x - \cos x = \frac{1}{4}\].
If \[\cos x + \sqrt{3} \sin x = 2,\text{ then }x =\]
The general value of x satisfying the equation
\[\sqrt{3} \sin x + \cos x = \sqrt{3}\]
If \[\cot x - \tan x = \sec x\], then, x is equal to
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
In (0, π), the number of solutions of the equation \[\tan x + \tan 2x + \tan 3x = \tan x \tan 2x \tan 3x\] is
The solution of the equation \[\cos^2 x + \sin x + 1 = 0\] lies in the interval
Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`
Solve the following equations:
sin θ + cos θ = `sqrt(2)`
Solve the following equations:
cos 2θ = `(sqrt(5) + 1)/4`
Solve `sqrt(3)` cos θ + sin θ = `sqrt(2)`
If a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a2 + b2 = m2 + n2
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.
