# How do you prove (sinx + cosx)(tanx + cotx)=secx + cscx?

May 28, 2018

$\textcolor{g r e e n}{\left(\sin x + \cos x\right) \left(\tan x + \cot x\right) = \sec x - \cos x + \cos x + \sin x + \csc x - \sin x = \sec x + \csc x}$

#### Explanation:

show below:

$\textcolor{b l u e}{\left(\sin x + \cos x\right) \left(\tan x + \cot x\right) = \sec x + \csc x}$

$L . H . S = \textcolor{b l u e}{\left(\sin x + \cos x\right) \left(\tan x + \cot x\right)} =$

$\sin x \cdot \tan x + \sin x \cdot \cot x + \cos x \cdot \tan x + \cos x \cdot \cot x =$

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

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

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

$\sec x - \cos x + \cos x + \sin x + \csc x - \sin x = \textcolor{b l u e}{\sec x + \csc x} = R . H . S$

$\textcolor{red}{\text{Useful Trigonometric Identities}}$

${\cos}^{2} \theta + {\sin}^{2} \theta = 1$

$1 + {\tan}^{2} \theta = {\sec}^{2} \theta$

$\sin 2 \theta = 2 \sin \theta \cos \theta$

$\cos 2 \theta = {\cos}^{2} \theta - {\sin}^{2} \theta = 2 {\cos}^{2} \theta - 1 = 1 - 2 {\sin}^{2} \theta$

${\cos}^{2} \theta = \frac{1}{2} \left(1 + \cos 2 \theta\right)$

${\sin}^{2} \theta = \frac{1}{2} \left(1 - \cos 2 \theta\right)$

$\tan x = \sin \frac{x}{\cos} x$

$\cot x = \cos \frac{x}{\sin} x$

$\frac{1}{\cos} x = \sec x$

$\frac{1}{\sin} x = \csc x$

May 28, 2018

#### Explanation:

We know that,

color(red)((1)tantheta=sintheta/costheta and cottheta=costheta/sintheta

color(blue)((2)sin^2theta+cos^2theta=1

color(violet)((3)1/sintheta=csctheta and 1/costheta=sectheta

We have to prove,

$\left(\sin x + \cos x\right) \left(\tan x + \cot x\right) = \sec x + \csc x$

We take Left Hand Side :

LHS=(sinx+cosx)(tanx+cotx)...tocolor(red)(Apply(1)

$L H S = \left(\sin x + \cos x\right) \left(\sin \frac{x}{\cos} x + \cos \frac{x}{\sin} x\right)$

$L H S = \left(\sin x + \cos x\right) \left(\frac{{\sin}^{2} x + {\cos}^{2} x}{\sin x \cos x}\right)$

LHS=(sinx+cosx)(1/(sinxcosx))...tocolor(blue)(Apply(2)

$L H S = \sin \frac{x}{\sin x \cos x} + \cos \frac{x}{\sin x \cos x}$

LHS=1/cosx+1/sinx...tocolor(violet)(Apply(3)

$L H S = \sec x + \csc x$

$L H S = R H S$