(Q3)Prove that(cosec A - sin A)(sec A - cos A)=
1
tan A + cot A

Answer:

L.H.S=(cosec A - sin A)(sec A - cos A)

[ cosec A =
1
sin A
; sec A =
1
cos A
]

=(
1
sin A
 - sin A)(
1
cos A
- cos A)

= (
1 - sin2A
sin A
)
1 - cos2A
cos A

[1 - sin2A= cos2A and 1-cos2A=2A]

=
cos2A
sin A
×
2A
cos A

cos A.A----------(1)

R.H.S=
1
tan A + cot A

=
1
sin A
A
 +
A
sin A

=
1
sin 2A + 2A
A.sin A

[sin 2A + 2A=1]

=
1
1
A . sin A

=1 ×
A . sin A
1

= A. sin A ------------(2)

From (1)and(2),we get
L.H.S=R.H.S

[ (cosec A - sin A)(sec A - cos A)=
1
tan A + cot A
]