csc2x=2cscsecx
To solve the equation csc(2x) = 2csc(x)sec(x), we can start by simplifying the right side using the reciprocal identities.
Recall that csc(x) is equal to 1/sin(x), and sec(x) is equal to 1/cos(x).
So, 2csc(x)sec(x) can be written as 2(1/sin(x))(1/cos(x)).
Now, let's simplify the left side of the equation, csc(2x).
Remember that csc(2x) is equal to 1/sin(2x).
Combining these simplifications, we have 1/sin(2x) = 2(1/sin(x))(1/cos(x)).
To simplify further, we need to use the double-angle identities for sine.
sin(2x) = 2sin(x)cos(x).
Substituting this back into the equation, we have 1/2sin(x)cos(x) = 2(1/sin(x))(1/cos(x)).
Now, we can simplify this equation by multiplying both sides by 2sin(x)cos(x).
This yields (2sin(x)cos(x))/(2sin(x)cos(x)) = 2(1/sin(x))(1/cos(x))(2sin(x)cos(x)).
The left side simplifies to 1, and the right side simplifies to 2.
So, we have 1 = 2.
However, 1 does not equal 2.
Therefore, the equation csc(2x) = 2csc(x)sec(x) has no solution.
In summary, we followed the steps of simplifying both sides of the equation using the definitions of csc(x), sec(x), and the double-angle identity for sine. We found that the equation leads to a contradiction, so it has no solution.