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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:

\displaystyle \begin{align} & \frac{\text{1-}\sin v}{\cos v}=\frac{\text{1-}\sin v}{\cos v}\centerdot \frac{1+\sin v}{1+\sin v}=\frac{1-\sin ^{2}v}{\cos v\left( 1+\sin v \right)} \\ & =\frac{\cos ^{2}v}{\cos v\left( 1+\sin v \right)}. \\ \end{align}

Eliminating \displaystyle \cos v then gives the answer:

\displaystyle \frac{\cos ^{2}v}{\cos v\left( 1+\sin v \right)}=\frac{\cos v}{1+\sin v}