Express sec2x in terms of tanx and secx
I know you have to
sec(2x) = 1/cos(2x) = 1/(cos²x - sin²x)
But how do you split that. Like how to simplify that?
sec²x=1+tan²x
To simplify the expression sec(2x), we can rewrite it in terms of tan(x) and sec(x) using trigonometric identities.
1. First, recall that tan(x) is equal to sin(x) / cos(x). Now, let's express sin²(x) in terms of tan(x) and sec(x).
sin²(x) = (sin(x))^2
= (sin(x))^2 * (sec(x))^2 / (sec(x))^2 [Multiply and divide by (sec(x))^2]
= (1 - cos²(x)) * (1 + tan²(x)) / (sec(x))^2 [Using the Pythagorean identity: sin²(x) = 1 - cos²(x) and tan²(x) = sin²(x) / cos²(x)]
2. Next, substitute the expression for sin²(x) into the original expression for sec(2x).
sec(2x) = 1 / (cos²(x) - sin²(x))
= 1 / (cos²(x) - (1 - cos²(x)) * (1 + tan²(x)) / (sec(x))^2)
= 1 / (cos²(x) - (1 - cos²(x)) * (1 + tan²(x)) * (1 / (sec(x))^2))
3. Simplify the expression further.
sec(2x) = 1 / (cos²(x) - (1 - cos²(x)) * (1 + tan²(x)) * (1 / (sec(x))^2))
= 1 / (cos²(x) - (1 - cos²(x)) * (1 + tan²(x)) / (cos(x))^2)
= 1 / (cos²(x) - (cos²(x) - cos⁴(x) + tan²(x) - cos²(x) * tan²(x)) / (cos(x))^2)
= 1 / (cos²(x) - cos²(x) + cos⁴(x) - tan²(x) + cos²(x)tan²(x)) / (cos(x))^2)
= 1 / (cos⁴(x) - tan²(x) + cos²(x)tan²(x)) / (cos(x))^2)
= 1 / (cos⁴(x) + cos²(x)tan²(x) - tan²(x)) / (cos(x))^2)
= (cos(x))^2 / (cos⁴(x) + cos²(x)tan²(x) - tan²(x))
So, sec(2x) can be expressed as (cos(x))^2 / (cos⁴(x) + cos²(x)tan²(x) - tan²(x)).
To simplify the expression sec(2x), you can use the identity cos(2x) = cos²x - sin²x.
Here's how you can simplify step by step:
1. Start with the expression sec(2x) = 1/cos(2x).
2. Substitute cos(2x) with the identity cos²x - sin²x: sec(2x) = 1/(cos²x - sin²x).
3. Factor the denominator: sec(2x) = 1/[(cosx + sinx)(cosx - sinx)].
4. Now, we can express the factors cosx + sinx and cosx - sinx in terms of tanx and secx.
To express cosx + sinx in terms of tanx, divide both sides of the equation by cosx:
(cosx + sinx)/cosx = 1 + tanx.
Therefore, cosx + sinx = cosx(1 + tanx).
To express cosx - sinx in terms of tanx, divide both sides of the equation by cosx:
(cosx - sinx)/cosx = 1 - tanx.
Therefore, cosx - sinx = cosx(1 - tanx).
5. Substitute the expressions for cosx + sinx and cosx - sinx into the original expression:
sec(2x) = 1/[(cosx(1 + tanx))(cosx(1 - tanx))].
6. Simplify the expression:
sec(2x) = 1/(cosx * cosx * (1 + tanx) * (1 - tanx)).
7. Combine like terms:
sec(2x) = 1/(cos²x * (1 - tan²x)).
8. Recall the identity tan²x = sec²x - 1:
sec(2x) = 1/(cos²x * (1 - (sec²x - 1))).
9. Simplify further:
sec(2x) = 1/(cos²x * (2 - sec²x)).
And there you have it! sec(2x) expressed in terms of tanx and secx is 1/(cos²x * (2 - sec²x)).
divide top and bottom by cos^2 (multiply by sec^2) and you have
sec^2(x)/(1-tan^2(x))