# Math Help - Prove 2 sec X = ... (Please help!)

1. ## Verify (1+sinx)/cosx + cosx/(1+sinx) = 2secx (Please help!)

Verify:

(1+sinx)/cosx + cosx/(1+sinx) = 2secx

Thanks.

2. Originally Posted by SMA777
Verify:

(1+sinx)/cosx + cosx/(1+sinx) = 2secx

Thanks.
$\frac{1 + \sin{x}}{\cos{x}} + \frac{\cos{x}}{1+\sin{x}}$

$\frac{(1+\sin^2{x}) + \cos^2{x}}{\cos{x}(1 + \sin{x})}$

$\frac{1 + 2\sin{x} +\sin^2{x} + \cos^2{x}}{\cos{x}(1 + \sin{x})}$

$\frac{1 + 2\sin{x} + 1}{\cos{x}(1 + \sin{x})}$

$\frac{2 + 2\sin{x}}{\cos{x}(1 + \sin{x})}$

$\frac{2(1 + \sin{x})}{\cos{x}(1 + \sin{x})}$

$\frac{2}{\cos{x}}$

$2\sec{x}$

3. $\frac{1+\sin \left( x \right)}{\cos \left( x \right)}+\frac{\cos \left( x \right)}{1+\sin \left( x \right)}=2\sec \left( x \right)$

$\frac{1+2\sin \left( x \right)+\sin ^{2}\left( x \right)+\cos ^{2}\left( x \right)}{\cos \left( x \right)\cdot 1+\sin \left( x \right)}=2\sec \left( x \right)
$

$\frac{2+2\sin \left( x \right)}{\cos \left( x \right)\cdot 1+\sin \left( x \right)}=2\sec \left( x \right)$

$\frac{2\cdot \left( 1+\sin \left( x \right) \right)}{\cos \left( x \right)\cdot 1+\sin \left( x \right)}=2\sec \left( x \right)$

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

$
2\cdot \frac{1}{\cos \left( x \right)}=2\sec \left( x \right)$

$
2\sec \left( x \right)=2\sec \left( x \right)$

4. Originally Posted by FrY
$\frac{2+2\sin \left( x \right)}{\cos \left( x \right)\cdot 1+\sin \left( x \right)}=2\sec \left( x \right)$

$\frac{2\cdot \left( 1+\sin \left( x \right) \right)}{\cos \left( x \right)\cdot 1+\sin \left( x \right)}=2\sec \left( x \right)$

Could someone please explain going from ${2+2\sin(x)}$ to ${2\cdot \left( 1+\sin \left( x \right) \right)}$ ?
I am unsure of how this happened.

5. Originally Posted by tiar
Could someone please explain going from ${2+2\sin(x)}$ to ${2\cdot \left( 1+\sin \left( x \right) \right)}$ ?
I am unsure of how this happened.
What we did is factorize the expression $2+2\sin \left( x \right)$ being $2$ the common factor.