Q.5) Prove the following identities where the angles involved
are acute angles for which the expressions are defined.
EXERCISE 8.4
(i)
(cosec θ
–
cot θ)
2
=
1
–
cos θ
1
+
cos θ
Proof:
L.H.S
=
(cosec θ – cot θ)2
=
1
sin θ
–
cos θ
sin θ
2
=
1
–
cos θ
sin θ
2
=
(1
–
cos θ)2
sin2θ
=
(1
–
cos θ)2
=
(1
–
cos θ)
(1
–
cos θ)
(1
–
cos θ)
(1
cos θ)
+
=
(1 – cos θ)
(1 + cos θ)
∴
1 – cos2θ
(cosec θ
–
cot θ)2
=
1
–
cos θ
1
+
cos θ
1 – cos2θ = sin2θ
L.H.S = R.H.S
∴