If
2
1−cosA
=x, then the value of x is:
- A\cos^2(A/2)
- B\sin(A/2)
- C\cos(A/2)
- D\sin^2(A/2)
Solution & Step-by-step Explanation
From standard trigonometric half-angle identities, we know that:
cos2θ=1−2sin
2
θ
Let 2θ=A, which implies θ=
2
A
. Substituting this into the identity gives:
cosA=1−2sin
2
(
2
A
)
Rearranging the terms to isolate the sine squared expression:
2sin
2
(
2
A
)=1−cosA
sin
2
(
2
A
)=
2
1−cosA
Given that x=
2
1−cosA
, we have:
x=sin
2
(
2
A
)
cos2θ=1−2sin
2
θ
Let 2θ=A, which implies θ=
2
A
. Substituting this into the identity gives:
cosA=1−2sin
2
(
2
A
)
Rearranging the terms to isolate the sine squared expression:
2sin
2
(
2
A
)=1−cosA
sin
2
(
2
A
)=
2
1−cosA
Given that x=
2
1−cosA
, we have:
x=sin
2
(
2
A
)