cos theta upon 1- sin theta + cos theta upon 1+ sin theta =
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If I understood corretcly (please format your questions!!!!), you want to prove that
cos
(
θ
)
1
−
sin
(
θ
)
=
1
+
sin
(
θ
)
cos
(
θ
)
This can be proven by cross-multiplication: multiply both sides by
cos
(
θ
)
(
1
−
sin
(
θ
)
)
, i.e. by both denominators to get
cos
2
(
θ
)
=
(
1
+
sin
(
θ
)
)
(
1
−
sin
(
θ
)
)
On the right hand side we have the expression
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
, so the expression becomes
cos
2
(
θ
)
=
1
−
sin
2
(
θ
)
Which is true, because it derives immediately from the fundamental trigonometric equation
cos
2
(
θ
)
+
sin
2
(
θ
)
=
1
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