If tan(
2
A
)=x, then the value of x is:
- AsinA
1−cosA
- BsinA
1+cosA
- CsinA
1−cosA
- DsinA
1+cosA
Solution & Step-by-step Explanation
We can use standard trigonometric half-angle identities to express tan(
2
A
).
We know that:
1−cosA=2sin
2
(
2
A
)
sinA=2sin(
2
A
)cos(
2
A
)
Dividing the first identity by the second identity:
sinA
1−cosA
=
2sin(
2
A
)cos(
2
A
)
2sin
2
(
2
A
)
sinA
1−cosA
=
cos(
2
A
)
sin(
2
A
)
=tan(
2
A
)
Since tan(
2
A
)=x, we have:
x=
sinA
1−cosA
2
A
).
We know that:
1−cosA=2sin
2
(
2
A
)
sinA=2sin(
2
A
)cos(
2
A
)
Dividing the first identity by the second identity:
sinA
1−cosA
=
2sin(
2
A
)cos(
2
A
)
2sin
2
(
2
A
)
sinA
1−cosA
=
cos(
2
A
)
sin(
2
A
)
=tan(
2
A
)
Since tan(
2
A
)=x, we have:
x=
sinA
1−cosA