From Förberedande kurs i matematik 1
Because
tanv=sinvcosv, the left-hand side can be written using
cosv
as the common denominator:
1cosv−tanv=1cosv−sinvcosv=cosv1-sinv
Now, we observe that if we multiply top and bottom by with
1+sinv, the denominator will contain the denominator of the right-hand side as a factor and, in addition, the numerator can be simplified to give
1−sin2v =cos2v, using the conjugate rule:
cosv1-sinv=cosv1-sinv
1+sinv1+sinv=1−sin2vcosv
1+sinv
=cos2vcosv
1+sinv
Eliminating
cosv
then gives the answer:
cos2vcosv
1+sinv
=cosv1+sinv