# Please how I can prove that ? Cos^2(t)=1/1+tan^2(t) Thanks

Mar 22, 2018

I think you mean "prove" not "improve". See below

#### Explanation:

Consider the RHS

$\frac{1}{1 + {\tan}^{2} \left(t\right)}$

$\tan \left(t\right) = \sin \frac{t}{\cos} \left(t\right)$

So, ${\tan}^{2} \left(t\right) = {\sin}^{2} \frac{t}{\cos} ^ 2 \left(t\right)$

So RHS is now:
1/( 1+ (sin^2(t)/cos^2(t))

$\frac{1}{\frac{{\cos}^{2} \left(t\right) + {\sin}^{2} \left(t\right)}{\cos} ^ 2 \left(t\right)}$

${\cos}^{2} \frac{t}{{\cos}^{2} \left(t\right) + {\sin}^{2} \left(t\right)}$

Now : ${\cos}^{2} \left(t\right) + {\sin}^{2} \left(t\right) = 1$

RHS is ${\cos}^{2} \left(t\right)$, same as LHS.

QED.

Mar 22, 2018

$\text{see explanation}$

#### Explanation:

$\text{to prove this is an identity either manipulate the left side}$
$\text{into the form of the right side or manipulate the right side}$
$\text{into the form of the left side}$

$\text{using the "color(blue)"trigonometric identities}$

â€¢color(white)(x)tanx=sinx/cosx" and "sin^2x+cos^2x=1

$\text{consider the right side}$

$\Rightarrow \frac{1}{1 + {\sin}^{2} \frac{t}{\cos} ^ 2 t}$

$= \frac{1}{\frac{{\cos}^{2} t + {\sin}^{2} t}{\cos} ^ 2 t}$

$= \frac{1}{\frac{1}{\cos} ^ 2 t}$

$= 1 \times {\cos}^{2} \frac{t}{1} = {\cos}^{2} t = \text{ left side hence proved}$