If 2csc
2
A=x, then the value of x is:
- AsecA−1
secA
+
secA+1
secA
- BsecA−1
cscA
+
secA+1
cscA
- CcscA−1
secA
+
cscA+1
secA
- DcscA−1
cscA
+
cscA+1
cscA
Solution & Step-by-step Explanation
Let's simplify the expression in Option A:
Expression=
secA−1
secA
+
secA+1
secA
Take secA as a common factor:
secA(
secA−1
1
+
secA+1
1
)
Find a common denominator:
=secA(
(secA−1)(secA+1)
(secA+1)+(secA−1)
)
=secA(
sec
2
A−1
2secA
)
Using the identity sec
2
A−1=tan
2
A:
=
tan
2
A
2sec
2
A
Now convert secA and tanA into sinA and cosA:
=
cos
2
A
sin
2
A
2⋅
cos
2
A
1
=
sin
2
A
2
=2csc
2
A
This perfectly matches the given value x=2csc
2
A.
Expression=
secA−1
secA
+
secA+1
secA
Take secA as a common factor:
secA(
secA−1
1
+
secA+1
1
)
Find a common denominator:
=secA(
(secA−1)(secA+1)
(secA+1)+(secA−1)
)
=secA(
sec
2
A−1
2secA
)
Using the identity sec
2
A−1=tan
2
A:
=
tan
2
A
2sec
2
A
Now convert secA and tanA into sinA and cosA:
=
cos
2
A
sin
2
A
2⋅
cos
2
A
1
=
sin
2
A
2
=2csc
2
A
This perfectly matches the given value x=2csc
2
A.