(Q5)prove that
1+tan
2
A
1+cot
2
A
= (
1+tan A
1+cot A
)
2
=tan
2
A
Answer:
L.H.S=
1+tan
2
A
1+cot
2
A
=
sec
2
A
cosec
2
A
[1+tan
2
A=sec
2
A and 1+cot
2
A=cosec
2
A]
[sec A=
1
A
and cosec A=
1
A
]
=
1
2
A
1
2
A
=
1
2
A
×
2
A
1
=
2
A
2
A
= tan
2
A =R.H.S
L.H.S=(
1+tan A
1+cot A
)
2
= (
1+
A
cos A
1+
A
sin A
)
2
= (
cos A +
A
cos A
sin A +
A
sin A
)
2
=
(cos A +
A)
2
cos
2
A
×
sin
2
A
(
A + sin A)
2
=
2
A
2
A
= tan
2
A = R.H.S
clear