If tan(
2
A
)=x, then the value of x is:
- A\sqrt{\frac{1+\cos A}{1-\cos A}}
- B\frac{1-\cos A}{1+\cos A}
- C\frac{1+\cos A}{1-\cos A}
- D\sqrt{\frac{1-\cos A}{1+\cos A}}
Solution & Step-by-step Explanation
Using half-angle trigonometric formulas:
sin
2
(
2
A
)=
2
1−cosA
cos
2
(
2
A
)=
2
1+cosA
Dividing the two equations:
tan
2
(
2
A
)=
cos
2
(
2
A
)
sin
2
(
2
A
)
=
1+cosA
1−cosA
Taking the square root on both sides:
tan(
2
A
)=
1+cosA
1−cosA
Therefore, x=
1+cosA
1−cosA
.
sin
2
(
2
A
)=
2
1−cosA
cos
2
(
2
A
)=
2
1+cosA
Dividing the two equations:
tan
2
(
2
A
)=
cos
2
(
2
A
)
sin
2
(
2
A
)
=
1+cosA
1−cosA
Taking the square root on both sides:
tan(
2
A
)=
1+cosA
1−cosA
Therefore, x=
1+cosA
1−cosA
.