Trigonometric Functions & Equations

The general solution of the equation $\sin ^{100} \mathrm{x}-\cos ^{100} \mathrm{x}=1$ is

$\theta$ elimination from the equation $x^2+y^2=\frac{x \cos 3 \theta+y \sin 3 \theta}{\cos ^3 \theta}=\frac{y \cos 3 \theta-x \sin 3 \theta}{\sin ^3 \theta}$ will be

Let $f_n(x)=\tan \frac{x}{2}(1+\sec x)(1+\sec 2 x) \ldots\left(1+\sec 2^n x\right)$, then

The solution set of the equation $\left(x \in\left(0, \frac{\pi}{2}\right)\right) \tan (\pi \tan x)=\cot (\pi \cot x)$, is

If $0 \leq a, b \leq 3$ and the equation $x^2+4+3 \cos (a x+b)=2 x$ has real solutions, then the value of $(a+b)$ is

If $\cos (\theta+\phi)=\frac{3}{5}$ and $\sin (\theta-\phi)=\frac{5}{13}, 0<\theta, \phi<\frac{\pi}{4}$, then $\cot (2 \theta)$ has the value

If the equation $\sin ^4 x-(p+2) \sin ^2 x-(p+3)=0$ has a solution, the $p$ must lie in the interval

If \(A\) and \(B\) are acute angles such that \(\sin A=\sin ^2 B\) and \(2 \cos ^2 A=3 \cos ^2 B\), then \((A, B)=\)

If \(0< \theta<\frac{\pi}{2}\) and \(\tan 3 \theta \neq 0\), then \(\tan \theta+\tan 2 \theta+\tan 3 \theta=0\) if \(\tan \theta \cdot \tan 2 \theta=\mathrm{k}\) where \(\mathrm{k}=\)

The expression \(\cos ^2 \phi+\cos ^2(\theta+\phi)-2 \cos \theta \cos \phi \cos (\theta+\phi)\) is

If \({1 \over 6}\sin \theta ,\cos \theta ,\tan \theta\) are in G.P, then the solution set of \(\theta\) is
(Here \(n \in N\))

If \((\cot {\alpha _1})(\cot {\alpha _2})\,......\,(\cot {\alpha _n}) = 1,0 < {\alpha _1},{\alpha _2},....\,{\alpha _n} < \pi /2\), then the maximum value of \((\cos {\alpha _1})(\cos {\alpha _2}).....(\cos {\alpha _n})\) is given by

The equation 6x + 8x = 10x has

\(\cos (2x + 7) = a(2 - \sin x)\) can have a real solution for

Let f(x) = sin x + cos ax be periodic function. Then,

If \({e^{\sin x}} - {e^{-\sin x}} - 4 = 0\), then the number of real values of x is

The graphs of the polynomial x2 \(-\) 1 and cos x intersect

The approximate value of sin31\(^\circ\) is

If sin6\(\theta\) + sin4\(\theta\) + sin2\(\theta\) = 0, then general value of \(\theta\) is

The equation \(\sin x(\sin x + \cos x) = k\) has real solutions, where k is a real number. Then,

If the matrix \(A = \left[ {\matrix{ 2 & 0 & 0 \cr 0 & 2 & 0 \cr 2 & 0 & 2 \cr } } \right]\), then \({A^n} = \left[ {\matrix{ a & 0 & 0 \cr 0 & a & 0 \cr b & 0 & a \cr } } \right],n \in N\), where

If the first and (2n \(-\) 1)th terms of an AP, GP and HP are equal and their nth terms are respectively a, b, c, then always

For non-zero vectors a and b, if | a + b | < | a \(-\) b |, then a and b are

The value of \(\cos 15^\circ \cos 7{{1^\circ } \over 2}\sin 7{{1^\circ } \over 2}\) is

If f(x) is an odd differentiable function defined on (\(-\)\(\infty\), \(\infty\)) such that f'(3) = 2, then f'(\(-\)3) is equal to

If the function f : R \(\to\) R is defined by f(x) = (x2 + 1)35, \(\forall\) x\(\in\)R, then f is

\(\int {{{\log \sqrt x } \over {3x}}} dx\) is equal to

The order of the differential equation of all parabolas whose axis of symmetry along X-axis is

The number of points at which the function f(x) = max {a \(-\) x, a + x, b}, \(-\) \(\infty\) < x < \(\infty\), 0 < a < b cannot be differentiable, is

The locus of the point of intersection of the straight lines \({x \over a} + {y \over b} = K\) and \({x \over a} - {y \over b} = {1 \over K}\), where K is a non-zero real variable, is given by

The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together, is

The points (\(-\)a, \(-\)b), (a, b), (0, 0) and (a2, ab), a \(\ne\) 0, b \(\ne\) 0 are always

If the parabola x2 = ay makes an intercept of length \(\sqrt {40}\) units on the line y \(-\) 2x = 1, then, a is equal to

The smallest positive root of the equation tan x \(-\) x = 0 lies in

Let S be the set of points, whose abscissae and ordinates are natural numbers. Let P \(\in\) S, such that the sum of the distance of P from (8, 0) and (0, 12) is minimum among all elements in S. Then, the number of such points P and S is

If z1, z2, z3 are imaginary numbers such that \(|{z_1}|\, = \,|{z_2}|\, = \,|{z_3}|\, = \,\left| {{1 \over {{z_1}}} + {1 \over {{z_2}}} + {1 \over {{z_3}}}} \right|\, = \,1\), then \(|{z_1} + {z_2} + {z_3}|\) is

A straight line joining the points (1, 1, 1) and (0, 0, 0) intersects the plane 2x + 2y + z = 10 at

The equation x3 \(-\) yx2 + x \(-\) y = 0 represents

The area enclosed by \(y = \sqrt {5 - {x^2}}\) and \(y = |x - 1|\) is

Let \(Q = \left[ {\matrix{ {\cos {\pi \over 4}} & { - \sin {\pi \over 4}} \cr {\sin {\pi \over 4}} & {\cos {\pi \over 4}} \cr } } \right]\) and $$x = \left[ {\matrix{
{{1 \over {\sqrt 2 }}} \cr
{{1 \over {\sqrt 2 }}}...

\(1 + {}^n{C_1}\cos \theta + {}^n{C_2}\cos 2\theta + ... + {}^n{C_n}\cos n\theta\) equals

If x is a positive real number different from 1 such that logax, logbx, logcx are in AP, then

Angle between the planes x + y + 2z = 6 and 2x \(-\) y + z = 9 is

The value of cos 15\(^\circ\) \(-\) sin 15\(^\circ\) is

If \(\sin \theta + \cos \theta = 0\) and \(0 < \theta < \pi\), then \(\theta\) =

If \(\theta\) + \(\phi\) = \(\pi\)/4, then (1 + tan \(\theta\)) (1 + tan \(\phi\)) is equal to

Let \(\tan \alpha = {a \over {a + 1}}\) and \(\tan \beta = {1 \over {2a + 1}}\) then \(\alpha + \beta\) is

The number of solutions of \(2\sin x + \cos x = 3\) is

If \(\sin \theta = {{2t} \over {1 + {t^2}}}\) and \(\theta\) lies in the second quadrant, then cos \(\theta\) is equal to

Find the values of x, (\(-\) \(\pi\), < x < \(\pi\), x \(\ne\) 0) satisfying the equation, \({8^{1 + |\cos x| + |{{\cos }^2}x| + ......\infty }} = {4^3}\).