Question

I need help with this practice problem solving I could not get the entire problem in one photo so I will send an additional photo of the rest

Answer 1

`(\cos x)/(1-\sin x)=(1)/(\cos x)+(\sin x)/(\cos x)`
`(\cos x)/(1-\sin x)=(1+\sin x)/(\cos x)`
`(\cos x)/(1-\sin x)=((1+\sin x)\cdot\cos x)/(1-\sin ^2x)`
`(\cos x)/(1-\sin x)=\frac{(1+\sin x)\cdot\cos x}{(1-\sin^{}x)(1+\sin x)}`
`(\cos x)/(1-\sin x)=(\cos x)/(1-\sin x)`

Step-by-step explanation:

First, we realise that

`\sec x=(1)/(\cos x)`

and

`\tan x=(\sin x)/(\cos x)`

therefore,

`\sec x+\tan x=(1)/(\cos x)+(\sin x)/(\cos x)`

Now,

`(1)/(\cos x)+(\sin x)/(\cos x)=(1+\sin x)/(\cos x)`

Using the fact that

`(1)/(\cos x)=(\cos x)/(\cos^2x)=(\cos x)/(1-\sin^2x)`

Therefore,

`(1+\sin x)/(\cos x)=(\cos x(1+\sin x))/(1-\sin^2x)`
`=(\cos x(1+\sin x))/((1-\sin x)(1+\sin x))=(\cos x)/(1-\sin x)`

Hence, we have shown that

`\boxed{\sec x+\tan x=(\cos x)/(1-\sin x)\text{.}}`