# Does 2+2tanx=2secx?

$2 + 2 \tan x \ne 2 \sec x$

#### Explanation:

2+2tanx=2secx ?

$2 + \frac{2 \sin x}{\cos} x = \frac{2}{\cos x}$

$\frac{2 \left(\cos x\right)}{\cos} x + \frac{2 \sin x}{\cos} x = \frac{2}{\cos x}$

$\frac{2 \cos x + 2 \sin x}{\cos} x = \frac{2}{\cos x}$

$\frac{2 \left(\cos x + \sin x\right)}{\cos} x = \frac{2}{\cos x}$

For this to be an identity, $\cos x + \sin x$ must equal 1 for all values of $x$. But, for instance, with $x = \frac{\pi}{2} \implies \sin x = \cos x = \frac{\sqrt{2}}{2}$.

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2} \ne 1$

And so the identity is not valid.

$2 + 2 \tan x \ne 2 \sec x$

May 4, 2017

Only when $x = 2 n \times {360}^{\circ}$, where $n$ is an integer.

#### Explanation:

No, if you mean $2 + 2 \tan x = 2 \sec x$ as an identity it is not.

But we can solve this for $x$ and at some values we may have

$2 + 2 \tan x = 2 \sec x$.

As $2 + 2 \tan x = 2 \sec x$

$1 + \tan x = \sec x$

and multiplying by $\cos x$ we get

$\cos x + \sin x = 1$

or $\sqrt{2} \left(\cos x \times \frac{1}{\sqrt{2}} + \sin x \times \frac{1}{\sqrt{2}}\right) = 1$

or $\cos x \times \frac{1}{\sqrt{2}} + \sin x \times \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

or $\cos x \cos {45}^{\circ} + \sin x \sin {45}^{\circ} = \cos {45}^{\circ}$

or $\cos \left(x - {45}^{\circ}\right) = \cos {45}^{\circ}$

i.e. $x - {45}^{\circ} = 2 n \times {360}^{\circ} \pm {45}^{\circ}$, where $n$ is an integer.

i.e. $x = 2 n \times {360}^{\circ} = {90}^{\circ}$ or $x = 2 n \times {360}^{\circ}$

But at first solution, both $\tan x$ and $\sec x$ are not defined

Hence $x = 2 n \times {360}^{\circ}$

graph{(y-secx+tanx+1)=0 [-10, 10, -5, 5]}