Prove 2 sec X = ... (Please help!)

**SMA777** · December 8th 2008, 02:24 PM

Verify:

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

Thanks.

**11rdc11** · December 8th 2008, 05:41 PM

Originally Posted by SMA777

Verify:

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

Thanks.

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

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

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

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

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

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

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

$2\sec{x}$

**FrY** · December 8th 2008, 05:41 PM

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

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

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

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

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

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

$<br /> 2\sec \left( x \right)=2\sec \left( x \right)$

**tiar** · December 9th 2008, 10:41 AM

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.

**FrY** · December 9th 2008, 12:39 PM

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.

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