Q. General solution of the equation $tan^{2}\theta +sec2\theta =1$ is

NTA AbhyasNTA Abhyas 2022 Report Error

A

$m\pi ,n\pi +\frac{\pi }{3}, \, m\in I, \, n\in I$

B

$m\pi ,n\pi \pm\frac{\pi }{3}, \, m\in I, \, n\in I$

C

$m\pi ,n\pi \pm\frac{\pi }{6}, \, m\in I, \, n\in I$

D

None of these

Solution:

Using, $sec2\theta =\frac{1}{c o s 2 \theta }=\frac{1 + t a n^{2} \theta }{1 - t a n^{2} \theta }$
$\Rightarrow $ We can write the given equation as $tan^{2}\theta +\frac{1 + t a n^{2} \theta }{1 - t a n^{2} \theta }=1$
$\Rightarrow $ $tan^{2}\theta \left(\right.1-tan^{2}\theta \left.\right)+1+tan^{2}\theta =1-tan^{2}\theta $
$\Rightarrow $ $3tan^{2}\theta -tan^{4}\theta =0\Rightarrow tan^{2}\theta \left(\right.3-tan^{2}\theta \left.\right)=0$
$\Rightarrow \tan \theta=0 \Rightarrow \theta=m \pi, m \in I$
$\Rightarrow \text{tan\theta } = 0 \Rightarrow \text{\theta } = m \text{\pi ,} \, m \in I$
or $tan^{2} \theta = tan^{2} \frac{\pi }{3}$ $\Rightarrow \theta =n\pi \pm\frac{\pi }{3}$ , $n \in I$

Questions from NTA Abhyas 2022

1. Let points $A_{1},A_{2}$ and $A_{3}$ lie on the parabola $y^{2}=8x.$ If $\Delta A_{1}A_{2}A_{3}$ is an equilateral triangle and normals at points $A_{1},A_{2}$ and $A_{3}$ on this parabola meet at the point $\left(h , 0\right)$ , then the value of $h$ is

2. The lines represented by the equation $x^{2}+2\sqrt{3}xy+3y^{2}-3x-3\sqrt{3}y-4=0$ , are

Straight Lines

3. If the roots of the equation $ax^{2}+bx+c=0$ are in the ratio $m:n$ , then

Complex Numbers and Quadratic Equations

4. If $a,b,c,d,e,f$ are in arithmetic progression. Then $e - c$ is equal to

Sequences and Series

5. The sum to $n$ terms of $\left[\frac{1}{1.3} + \frac{2}{1.3 . 5} + \frac{3}{1.3 . 5.7} + \frac{4}{1.3 . 5.7 . 9} + . . . . . . . .\right]$ is

Mathematics Most Viewed Questions

1. The solution of $\frac{dy}{dx} = \frac{y}{x}+\tan \frac{y}{x}$ is

WBJEE 2011 Differential Equations

2. The solution of the differential equation $\frac{dy}{dx} = (x +y)^2$ is

COMEDK 2009 Differential Equations

3. $\int\frac{1}{\sin x\, \cos x}$ dx is equal to

KEAM 2016 Integrals

4. If $\begin{bmatrix}1&- \tan\theta \\ \tan \theta&1\end{bmatrix}\begin{bmatrix}1&\tan \theta \\ - \tan \theta &1\end{bmatrix}^{-1} = \begin{bmatrix}a&-b\\ b&a\end{bmatrix}$ then

COMEDK 2009 Matrices

5. The value of $ \int{\frac{{{x}^{2}}+1}{{{x}^{4}}-{{x}^{2}}+1}}dx $ is

KEAM 2007 Integrals