v)
cos A – sin A + 1
cos A + sin A – 1
=
cosec A + cot A,
Proof:
L.H.S
=
cos A – sin A + 1
cos A + sin A – 1
=
cot A
–
1
cosec A
+
cot A
1
cosec A
–
+
cot A
cosec A
+
=
(
)
–
(
1
+
cot A
–
cosec A
)
cos A
–
sin A
cos A
+
sin A
–
+
1
=
sin A
sin A
sin A
sin A
sin A
(cosec2A – cot2A)
1
EXERCISE 8.4
Q.5) Prove the following identities where the angles involved
are acute angles for which the expressions are defined.
sin A
using cosec² A = 1 + cot² A.
cosec2θ – cot2θ = 1
[Dividing numerator and denominator by sin A]
v)
cos A – sin A + 1
cos A + sin A – 1
= cosec A + cot A
EXERCISE 8.4
Q.5) Prove the following identities where the angles involved
are acute angles for which the expressions are defined.
using cosec² A = 1 + cot² A.
=
cot A
cosec A
+
(
)
–
cosec A
cot A
+
(
)
cosec A
cot A
–
(
)
(
1
+
cot A
–
cosec A
)
=
cotA
cosecA
+
(
)
(
1
–
cosecA
+
cotA
)
(
1
+
cot A
–
cosec A
)
=
cot A
+
cosec A
∴
L.H.S. = R.H.S.
cos A – sin A + 1
cos A + sin A – 1
= cosec A + cot A
Proof:
cot A
cosec A
+
=
(
)
–
(
)
(
1
+
cot A
–
cosec A
)
cosec2A – cot2A
a2 – b2 = (a + b) (a – b)
=
cosec A
cot A
+
(
)
[1 –
–
(cosec A
cotA)]
1
+
cot A
–
cosec A
∴