2x=sec theta and 2/x=tan theta find the value of 2(x^2 - 1/x^2)
sec^2 = 1+tan^2 so
(2x)^2 = 1+(2/x)^2
4x^2 = 1 + 4/x^2
4x^4 - x^2 - 4 = 0
and proceed as above.
Or, forget about the value of x altogether.
secØ = 2x, so x = secØ/2
tanØ = 2/x, so 1/x = tanØ/2
2(x^2-1/x^2)
2(sec^2Ø/4 - tan^2Ø/4)
1/2(sec^2Ø - tan^2Ø)
1/2
Using the definitions of tan and sec in terms of
opposite, adjacent and hypotenuse ...
tanØ = 2/x = opp/adj
so we have a right-angled triangle with base angle Ø, opposite as 2 and adjacent as x
by Pythagoras,
r , (the hypotenuse) = (x^2 + 4)^(1/2 or √(x^2 + 4)
and secØ = √(x^2 + 4)/x
but we are given that secØ = 2x , so ....
2x = √(x^2 + 4)/x
2x^2 = √(x^2 + 4)
square both sides:
4x^4 = x^2 + 4
4x^4 - x^2 - 4 = 0
x^2 = (1 ± √(1+64)/8
x^2 = (1 + √65)/8 , rejecting the negative value since x^2 can't be negative
so 2(x^2 - 1/x^2)
= 2( (1+√65)/8) - (8/(1+√65) )
= 0.5
To find the value of 2(x^2 - 1/x^2), we need to first solve the given equations and find the value of x and theta.
Let's start with the equation 2x = sec(theta).
Using the identity sec(theta) = 1/cos(theta), we can rewrite the equation as:
2x = 1/cos(theta)
Multiplying both sides by cos(theta), we get:
2x * cos(theta) = 1
Next, let's solve the equation 2/x = tan(theta).
Using the identity tan(theta) = sin(theta)/cos(theta), we can rewrite the equation as:
2/x = sin(theta)/cos(theta)
Multiplying both sides by x and cos(theta), we get:
2 * cos(theta) = x * sin(theta)
Now, we have a system of equations:
2x * cos(theta) = 1
2 * cos(theta) = x * sin(theta)
We can solve this system of equations simultaneously using trigonometric identities.
Dividing the second equation by the first equation, we get:
(2 * cos(theta)) / (2x * cos(theta)) = (x * sin(theta)) / 1
Simplifying the equation, we get:
1/x = sin(theta)
Now, we have two equations:
2x * cos(theta) = 1
1/x = sin(theta)
From the second equation, we can substitute sin(theta) with 1/x in the first equation:
2x * cos(theta) = 1
1/x = 1/x
This implies that 2x * cos(theta) = 1.
Simplifying further, we get:
2x = 1/cos(theta)
2x = sec(theta)
We notice that this equation is the same as the given equation 2x = sec(theta).
Therefore, the equations are consistent, and we can solve them together.
Since we have 2x = sec(theta), we can substitute sec(theta) with 2x in the expression to find the value of 2(x^2 - 1/x^2):
2(x^2 - 1/x^2) = 2[(2x)^2 - 1/(2x)^2]
= 2(4x^2 - 1/4x^2)
= 8x^2 - 1/2x^2
Now, we need to find the value of x to substitute it into the expression.
To do that, we go back to the equations 2x * cos(theta) = 1 and 1/x = sin(theta).
We can rewrite the first equation as:
2x = 1 / cos(theta)
x = 1 / (2cos(theta))
Substituting sin(theta) = 1/x in the second equation, we get:
1 / (2cos(theta)) = sin(theta)
1 / (2cos(theta)) = sin(theta) / 1
1 = 2cos^2(theta) * sin(theta)
Now, we need to find the value of cos^2(theta) * sin(theta).
Let's rewrite cos^2(theta) as (1 - sin^2(theta)):
1 = 2(1 - sin^2(theta)) * sin(theta)
1 = 2sin(theta) - 2sin^3(theta)
This is a cubic equation in terms of sin(theta).
By solving this equation, we can find the value of sin(theta).
Once we find the value of sin(theta), we can substitute it back into the expression 8x^2 - 1/2x^2 to find the final value of 2(x^2 - 1/x^2).