Question

How do you prove `(sin+cos)(tan+cot)=sec+csc`?

Answer 1

See explanation.

Explanation:

Prove: `(sinx+cosx)(tanx+cotx)=secx+cscx`

The change I made in each step is colored red.

`[1]color(white)(XX)(sinx+cosx)(tanx+cotx)`

`[2]color(white)(XX)=(sinx+cosx)(color(red)(sinx/cosx)+color(red)(cosx/sinx))`

`[3]color(white)(XX)=color(red)(sin^2x/cosx+cosx+cos^2x/sinx+sinx)`

`[4]color(white)(XX)=(sin^2x+color(red)(cos^2x))/cosx+(cos^2x+color(red)(sin^2x))/sinx`

`[5]color(white)(XX)=(color(red)(1))/cosx+(color(red)(1))/sinx`

`[6]color(white)(XX)=color(red)(secx)+color(red)(cscx)`