Question

Prove: cos⁡(x)/(1+sin⁡(x) )+(1+sin⁡(x))/cos⁡(x) =2sec⁡(x)

Answer 1

`(\cos x)/(1+\sin x)+(1+\sin x)/(\cos x)=2\sec x\\\\L_s=(\cos x\cos x)/(\cos x(1+\sin x))+((1+\sin x)(1+\sin x))/(\cos x(1+\sin x))\\\\=(\cos^2x+1+\sin x+\sin x+\sin^2x)/(\cos x(1+\sin x))=((\cos^2x+\sin^2x)+1+2\sin x)/(\cos x(1+\sin x))\\\\=(1+1+2\sin x)/(\cos x(1+\sin x))=(2+2\sin x)/(\cos x(1+\sin x))=(2(1+\sin x))/(\cos x(1+\sin x))\\\\=(2)/(\cos x)=2\cdot(1)/(\cos x)=2\sec x=R_s`


`\text{Used:}\\\\\sin^2\alpha+\cos^2\alpha=1\\\\\sec\alpha=(1)/(\cos\alpha)`

Answer 2

cos⁡(x)/(1+sin⁡(x) )+(1+sin⁡(x))/cos⁡(x) =2sec⁡(x)

Work on the left hand side.
[Common denominator is (1+sin(x))*cos(x)]
cos⁡(x)/(1+sin⁡(x) )+(1+sin⁡(x))/cos⁡(x)
= (cos(x)^2+(1+sin(x))^2)/((1+sin(x))*cos(x))
=(cos(x)^2+1+sin(x)^2+2sin(x))/((1+sin(x))*cos(x))
=(cos(x)^2+sin(x)^2+1+2sin(x))/((1+sin(x))*cos(x))
=(1+1+2sin(x))/((1+sin(x))*cos(x))
=(2+2sin(x))/((1+sin(x))*cos(x))
=2(1+sin(x))/((1+sin(x))*cos(x))
=2/cos(x)
=2 sec(x) [QED]