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

1 Answer
Oct 17, 2015

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)