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

1 Answer
Oct 17, 2015

See explanation.

Explanation:

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

The change I made in each step is colored red.

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

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

[3]color(white)(XX)=color(red)(sin^2x/cosx+cosx+cos^2x/sinx+sinx)[3]XX=sin2xcosx+cosx+cos2xsinx+sinx

[4]color(white)(XX)=(sin^2x+color(red)(cos^2x))/cosx+(cos^2x+color(red)(sin^2x))/sinx[4]XX=sin2x+cos2xcosx+cos2x+sin2xsinx

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

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