TRIG Identities, Gr. 12:
Prove:
cot4x=1-tan^2x/2tan2x
just apply the double-angle formula for tan(2x) and take the reciprocal
okay, thank you, would that still apply if it was cot4x=1-tan^2(2x)/2tan2x? because ive got the right side for this one but the cot4x is throwing me off, how would you solve that side?
you are exactly right
tan(4x) = 2tan2x/(1-tan^2(2x))
so, cot(4x) is the reciprocal, as needed.
To prove the given trigonometric identity: cot(4x) = (1 - tan^2(x))/(2tan(2x)), we will start by manipulating the right-hand side (RHS) of the equation and simplify it step by step until it is equal to the left-hand side (LHS).
Starting with the RHS: (1 - tan^2(x))/(2tan(2x))
1. Rewrite tan(2x) as sin(2x)/cos(2x):
(1 - tan^2(x))/(2(sin(2x)/cos(2x)))
2. Simplify by multiplying through by cos(2x) to get rid of the denominator:
(1 - tan^2(x))/(2sin(2x))
3. Rewrite tan^2(x) as (sin^2(x))/(cos^2(x)):
(1 - (sin^2(x))/(cos^2(x)))/(2sin(2x))
4. Combine the terms in the numerator:
(cos^2(x) - sin^2(x))/(cos^2(x) * 2sin(2x))
5. Keep in mind the following trigonometric identities:
- cos^2(x) - sin^2(x) = cos(2x)
- sin(2x) = 2sin(x)cos(x)
Applying these identities, we have:
cos(2x)/(2sin(2x) * cos^2(x))
6. Cancel out the 2 in the numerator and the denominator:
cos(2x)/(sin(2x) * cos^2(x))
7. Rewrite sin(2x) as 2sin(x)cos(x):
cos(2x)/(2sin(x)cos(x) * cos^2(x))
8. Cancel out sin(x) and cos(x) terms:
cos(2x)/(2cos^3(x))
9. Rewrite cos^3(x) as (cos(x))^3:
cos(2x)/(2(cos(x))^3)
10. Rewrite cos(2x) as cos^2(x) - sin^2(x):
(cos^2(x) - sin^2(x))/(2(cos(x))^3)
11. Apply the identity sin^2(x) = 1 - cos^2(x):
(cos^2(x) - (1 - cos^2(x)))/(2(cos(x))^3)
12. Simplify the numerator:
(cos^2(x) - 1 + cos^2(x))/(2(cos(x))^3)
13. Combine like terms:
(2cos^2(x) - 1)/(2(cos(x))^3)
14. Divide both the numerator and denominator by 2:
(cos^2(x) - 1)/(cos(x))^3
15. Apply the identity cos^2(x) - 1 = -sin^2(x):
(-sin^2(x))/(cos(x))^3
16. Rewrite -sin^2(x) as -1 * sin^2(x):
-sin^2(x)/(cos(x))^3
17. Divide numerator and denominator by -sin(x):
-sin(x)/(cos(x))^2
Finally, we obtain the left-hand side (LHS):
cot(4x) = -sin(x)/(cos(x))^2
Therefore, by manipulating the right-hand side (RHS) step by step, we have successfully proven that cot(4x) = (1 - tan^2(x))/(2tan(2x)).