Question

if tan theta=cot theta then the value of 2 tan theta+cos 2 theta is

Answer 1

Step-by-step explanation:

tan theta =cos theta

then

2 tan theta +2 cos theta = 2 cos theta + 2 sin theta

2 tan theta =2 sin theta

But I am not sure.

Answer 2

Given,

$\[\tan \theta = cot\theta \]$

Solution,

Know that, $\[\tan \theta = cot\theta \]$

Rewrite it,

$\[ \Rightarrow \tan \theta = \frac{1}{{\tan \theta }}\]$

$\[\begin{array}{l} \Rightarrow {\tan ^2}\theta = 1\\ \Rightarrow \tan \theta = \pm 1\\ \Rightarrow \theta = 45^\circ ,135^\circ \end{array}\]$

Calculate the value,

Consider $\[\theta = 45^\circ \]$

$\[ = 2\tan \theta + {\cos ^2}\theta \]$

$\[ = 2\tan 45^\circ + {\cos ^2}45^\circ \]$

$\[ = 2 \times 1 + {\left( {\frac{1}{{\sqrt 2 }}} \right)^2}\]$

$\[ = 2 + \frac{1}{2}\]$

$\[ = \frac{5}{2}\]$

Consider $\[\theta = 135^\circ \]$

$\[\begin{array}{l} = 2\tan \theta + {\cos ^2}\theta \\ = 2\tan 135^\circ + {\cos ^2}135^\circ \end{array}\]$

$\[ = 2 \times \left( { - 1} \right) + {\left( { - \frac{1}{{\sqrt 2 }}} \right)^2}\]$

$\[\begin{array}{l} = - 2 + \frac{1}{2}\\ = \frac{{ - 3}}{2}\end{array}\]$

Hence the values are $\[\frac{5}{2}\]$ and $\[\frac{{ - 3}}{2}\]$.