sin theta / 1 + cos theta + 1 + cos theta / sin theta is equals to 2 + 2 cos square theta
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Explanation:
sinθ×sinθ(1+cosθ)×sinθ
+
(
1
+
cos
θ)×(1+
cosθ
)sinθ×(1+cosθ)
= sin2θ
+
1
+
2
cos
θ
+
cos
2
θ
sin
θ
×
(
1
+
cos
θ
)
Using the pythagorean identity
cos
2
θ
+
sin
2
θ
=
1
=
2
+
2
cos
θ
sin
θ
×
(
1
+
cos
θ
)
=
2
(
1
+
cos
θ
)
sin
θ
×
(
1
+
cos
θ
)
=
2
sin
θ
Now recall that
1
sin
θ
=
csc
θ
=
2
csc
θ
Hopefully this helps!