Q.5) Prove the following identities where the angles involved
are acute angles for which the expressions are defined.
EXERCISE 8.4
(vi)
= sec A + tan A
1 + sin A
1 – sin A
Proof:
R.H.S
=
sec A
+
1
cos A
sin A
cos A
=
+
=
sin A
cos A
1
tan A
+
cos2θ = 1 – sin2θ
R.H.S
∴
… (i)
sin A
1
1
=
L.H.S.
+
–
sin A
=
sin A)
(1
– sin A)
sin A)
(1
+ sin A)
+
+
(1
(1
=
sin A)2
(1
1 – sin2A
+
=
sin A)2
(1
cos2A
+
∴ L.H.S
… (ii)
∴ L.H.S = R.H.S
[From (i) & (ii)]
∴
= sec A + tan A
1 + sin A
1 – sin A