Take next the cosine.
Let
y
=
cos
θ
{\displaystyle y=\cos \theta }
.
Now
cos
θ
=
sin
(
π
2
−
θ
)
{\displaystyle \cos \theta =\sin \left({\dfrac {\pi }{2}}-\theta \right)}
.
Therefore
d
y
=
d
(
sin
(
π
2
−
θ
)
)
=
cos
(
π
2
−
θ
)
×
d
(
−
θ
)
,
=
cos
(
π
2
−
θ
)
×
(
−
d
θ
)
,
d
y
d
θ
=
−
cos
(
π
2
−
θ
)
.
{\displaystyle {\begin{aligned}&{\begin{aligned}dy=d\left(\sin \left({\frac {\pi }{2}}-\theta \right)\right)&=\cos \left({\frac {\pi }{2}}-\theta \right)\times d(-\theta ),\\&=\cos \left({\frac {\pi }{2}}-\theta \right)\times (-d\theta ),\end{aligned}}\\&{\frac {dy}{d\theta }}=-\cos \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}}
And it follows that
d
y
d
θ
=
−
sin
θ
{\displaystyle {\frac {dy}{d\theta }}=-\sin \theta }
.
Lastly, take the tangent.
Let
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikisource.org/v1/":): {\displaystyle \begin{align} y &= \tan \theta, \\ dy &= \tan(\theta + d\theta) - \tan\theta. \\ \end{align}}
Expanding, as shown in books on trigonometry,
tan
(
θ
+
d
θ
)
=
tan
θ
+
tan
d
θ
1
−
tan
θ
⋅
tan
d
θ
;
whence
d
y
=
tan
θ
+
tan
d
θ
1
−
tan
θ
⋅
tan
d
θ
−
tan
θ
=
(
1
+
tan
2
θ
)
tan
d
θ
1
−
tan
θ
⋅
tan
d
θ
.
{\displaystyle {\begin{aligned}\tan(\theta +d\theta )&={\frac {\tan \theta +\tan d\theta }{1-\tan \theta \cdot \tan d\theta }};\\{\text{whence}}\quad \quad \quad dy&={\frac {\tan \theta +\tan d\theta }{1-\tan \theta \cdot \tan d\theta }}-\tan \theta \\&={\frac {(1+\tan ^{2}\theta )\tan d\theta }{1-\tan \theta \cdot \tan d\theta }}.\end{aligned}}}