simplifying into sine and cosine

[stacylou2000]

how do I simplify (sec x+csc x)(cos x-sin x)

 

[DrPhil]

how do I simplify (sec x + csc x)(cos x - sin x)

Write the first factor as (1/cosx + 1/sinx) and multiply the two binomials (FOIL). Look for double-angle formulas.

If you need more help, show us your work.

 

[Jamest]

Given (sec x+csc x)(cos x-sin x)
= secx.xosx-secxsinx+cscx.cosx-cscx.sinx
=1/cosx.cosx-1/cosx.sinx + 1/sinx.cosx – 1/sinx.sinx
= 0-tanx+cotx+0 // tanx = sinx/cosx , cotx = cosx/sinx //
=-tanx+cotx
​

 

[HallsofIvy]

That looks to me like the hard way to do it! The first thing I would do is replace sec(x) with 1/cos(x) and csc(x) with 1/sin(x). $\displaystyle (sec(x)+ csc(x))(cos(x)- sin(x))= (\frac{1}{cos(x)}+ \frac{1}{sin(x)})(cos(x)- sin(x))$
and now multiplying simplifies a lot.

Of course, if the object of the problem is "simplifying into sine and cosine", writing the final answer as "tan(x)+ cot(x)" can't be right.