From Förberedande kurs i matematik 1
Because tanv=sinvcosv, the left-hand side can be written using cosv as the common denominator,
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| 1cosv−tanv=1cosv−sinvcosv=cosv1−sinv.
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Now, we observe that if we multiply top and bottom 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 difference of two squares,
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| cosv1−sinv=cosv1−sinv 1+sinv1+sinv=1−sin2vcosv(1+sinv)=cos2vcosv(1+sinv). |
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Eliminating cosv then gives the answer,
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| cos2vcosv(1+sinv)=cosv1+sinv.
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