Trigonometric Equations

What is the general solution of $\cos ^{100} x-\sin ^{100} x=1$ ? where $n$ is an integer?

Consider the equation: $k \sin x+\cos 2 x=2 k-7$ If the equation possesses solution, then what is the maximum value of k ?

If $x=y \cos \left(\frac{2 \pi}{3}\right)=z \cos \left(\frac{4 \pi}{3}\right)$, then what is $x y+y z+$ $z x$ equal to?

The set of values of $\theta$ satisfying the inequation $2 \sin ^2 \theta-5 \sin \theta+2>0$, where $0 < \theta < 2 \pi$, is

Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) The number of elements in the set $\{x \in [-\pi, \pi] : \sin^6 x + \cos^4 x = 1\}$ (1) is $1$ (Q) The number…

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$ Then which of…

Let $\frac{\pi}{2} < x < \pi$ be such that $\cot x=\frac{-5}{\sqrt{11}}$. Then $\left(\sin \frac{11 x}{2}\right)(\sin 6 x-\cos 6 x)+\left(\cos \frac{11 x}{2}\right)(\sin 6 x+\cos 6 x)$ is equal to

For any $y\inℝ$, let $cot^{-1}(y)\in(0,\pi)$ and $tan^{-1}(y)\in(-\frac{\pi}{2},\frac{\pi}{2})$. Then the sum of all the solutions of the equation…

Consider the following lists: List-$I$ List-$II$ $(I)$ ${x\in[-\frac{2\pi}{3},\frac{2\pi}{3}]:cosx+sinx=1}$ $(P)$ has two elements $(II)$ ${x\in[-\frac{5\pi}{18},\frac{5\pi}{18}]:\sqrt{3}tan3x=1}$ $(Q)$ has three…

Let $|M|$ denote the determinant of a square matrix $M$. Let $g:[0,\frac{\pi}{2}]\toℝ$ be the function defined by $g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f(\frac{\pi}{2}-\theta)-1}$ where $f(\theta)=\frac{1}{2}|1 &…

Let $f(x)=sin⁡(\picos⁡x)$ and $g(x)=cos⁡(2\pisin⁡x)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order: $X={x:f(x)=0},Y={x:f^{'}(x)=0}$…

Let $f(x)=sin⁡(\picos⁡x)$ and $g(x)=cos⁡(2\pisin⁡x)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order: $X={x:f(x)=0},Y={x:f^{'}(x)=0}$…