(1/tan theta +cot theta)(1/1+sina)+1/1-sina=2sec^2a
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LHS :
1−cosθ
tanθ
+
1−tanθ
cotθ
=
sinθ
sinθ−cosθ
cosθ
sinθ
+
cosθ
cosθ−sinθ
sinθ
cosθ
=
sinθ−cosθ
1
[
cosθ
sin
2
θ
−
sinθ
cos
2
θ
]
=[
sinθ−cosθ
1
][
sinθcosθ
sin
3
θ−cos
3
θ
]
=
sinθcosθ
sin
2
θ+cos
2
θ+sinθcosθ
=
sinθcosθ
1+sinθcosθ
=secθcosec θ+1
=RHS
(iv)
secA
1+secA
=
cosA
1
1+
cosA
1
=
1
cosA+1
=
1−cosA
(cosA+1)(1−cosA)
=
1−cosA
sin
2
A
=RHS
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