how would you create an equation for sec(2x) using both sec(x) and csc(x)?
the steps i have so far are
1/(cos^2(x)-sin^2(x))
= 1/(1-1)/(sec^2(x)-csc^2(x))
but then i do not know what to do after this.
To create an equation for sec(2x) using both sec(x) and csc(x), we can start with the identity:
1 - cos^2(x) = sin^2(x)
Rearranging this identity, we get:
cos^2(x) = 1 - sin^2(x)
Now, let's express sec(2x) using sec(x) and csc(x).
We have the double angle identity:
sec(2x) = 1/cos(2x)
Next, we can express cos(2x) using the double angle identity for cosine:
cos(2x) = cos^2(x) - sin^2(x)
Substituting the previously rearranged identity for cos^2(x):
cos(2x) = 1 - sin^2(x) - sin^2(x)
cos(2x) = 1 - 2sin^2(x)
Finally, substitute this expression of cos(2x) back into the equation for sec(2x):
sec(2x) = 1/cos(2x) = 1 / (1 - 2sin^2(x))
So, the equation for sec(2x) using both sec(x) and csc(x) is:
sec(2x) = 1 / (1 - 2sin^2(x))