Lösung 4.3:8c

Aus Online Mathematik Brückenkurs 1

One could write tan2u as a quotient involving sine and cosine, and then continue with the formula for half-angles,

tan2u=sin2ucos2u=

but because this leads to square roots and difficulties with keeping a check on the correct sign in front of the roots, it is perhaps simpler instead to go backwards and work with the right-hand side.

We write u as 22u  and use the formula for double angles (so as to end up with a right-hand side which has 2u as its argument)

sinu1+cosu=sin22u1+cos22u=2cos2usin2u1+cos22u−sin22u

Writing the 1 in the denominator as cos22u+sin22u using the Pythagorean identity,

2cos2usin2u1+cos22u−sin22u=2cos2usin2ucos22u+sin22u+cos22u−sin22u=2cos22u2cos2usin2u=sin2ucos2u=tan2u