## anonymous one year ago verify cos(x)/(1+sin(x))+(1+sin(x))/cos(x)=2sec(x)

1. triciaal

|dw:1437698487751:dw|

2. DecentNabeel

manipulate the left side $\frac{\cos \left(x\right)}{1+\sin \left(x\right)}+\frac{1+\sin \left(x\right)}{\cos \left(x\right)}$ $\mathrm{Simplify}\:\frac{1+\sin \left(x\right)}{\cos \left(x\right)}+\frac{\cos \left(x\right)}{1+\sin \left(x\right)}:\quad \frac{\left(\sin \left(x\right)+1\right)^2+\cos ^2\left(x\right)}{\cos \left(x\right)\left(\sin \left(x\right)+1\right)}$ $=\frac{\left(1+\sin \left(x\right)\right)^2+\cos ^2\left(x\right)}{\left(1+\sin \left(x\right)\right)\cos \left(x\right)}$ $\mathrm{Use\:the\:following\:identity}:\quad \cos ^2\left(x\right)=1-\sin ^2\left(x\right)$ $=\frac{\left(1+\sin \left(x\right)\right)^2+1-\sin ^2\left(x\right)}{\left(1+\sin \left(x\right)\right)\cos \left(x\right)}$ $\mathrm{Simplify}\:\frac{\left(1+\sin \left(x\right)\right)^2+1-\sin ^2\left(x\right)}{\left(1+\sin \left(x\right)\right)\cos \left(x\right)}:\quad \frac{2}{\cos \left(x\right)}$ $\mathrm{Use\:the\:following\:identity:}\:\cos \left(x\right)=\frac{1}{\sec \left(x\right)}$ $=\frac{2}{\frac{1}{\sec \left(x\right)}}$ $=2\sec \left(x\right)$

3. DecentNabeel

are you understand @tamez1210

4. anonymous

Yes I do thank you!

5. DecentNabeel

wellcome :)