Q.5) Prove the following identities where the angles involved
are acute angles for which the expressions are defined.
EXERCISE 8.4
(vii)
sin θ
–
2 sin3θ
2cos3θ
–
cos θ
=
tan θ
Proof:
L.H.S
=
sin θ – 2 sin3θ
=
sin θ
(1
–
2 sin2θ)
cos θ
–
1)
=
sin θ
(1
–
2 sin2θ)
cos θ
[
2
(1
–
1]
=
sin θ
(1
–
2 sin2θ)
cos θ
(2
–
2 sin2θ
–
1)
=
sin θ
(1
–
2 sin2θ)
cos θ
(1
–
2 sin2θ)
=
sin θ
cos θ
=
tan θ
2 cos3θ – cos θ
(2cos2θ
– sin2θ)
L.H.S = R.H.S
∴
sin θ
–
2sin3θ
2cos3θ
–
cos θ
=
tan θ
∴
L.H.S
∴