If x=2sec (theta), express tan(theta) in terms of x.
To express tan(theta) in terms of x, we first need to find a trigonometric relationship between sec(theta) and tan(theta).
We know that sec(theta) is the reciprocal of cos(theta): sec(theta) = 1/cos(theta). And we also know that tan(theta) is the ratio of sin(theta) to cos(theta): tan(theta) = sin(theta)/cos(theta).
To relate sec(theta) and tan(theta), we can use the Pythagorean identity for trigonometric functions: sin^2(theta) + cos^2(theta) = 1.
Let's start by finding sin(theta). Rearranging the Pythagorean identity, we have sin^2(theta) = 1 - cos^2(theta). Taking the square root of both sides gives sin(theta) = √(1 - cos^2(theta)).
Now, let's substitute this value into the expression for tan(theta): tan(theta) = sin(theta)/cos(theta) = (√(1 - cos^2(theta)))/cos(theta).
Finally, we substitute x = 2sec(theta) into the expression for sec(theta): x = 2sec(theta) = 2/(cos(theta)). Therefore, cos(theta) = 2/x.
Substituting this value into the expression for tan(theta), we get: tan(theta) = (√(1 - (2/x)^2))/(2/x).
Simplifying further, we have: tan(theta) = √(1 - 4/x^2) * (x/2).
Hence, tan(theta) in terms of x is given by tan(theta) = √(1 - 4/x^2) * (x/2).
To express tan(theta) in terms of x, we need to use the trigonometric identity relating sec(theta) and tan(theta).
The identity is:
sec^2(theta) = 1 + tan^2(theta)
Let's rearrange this formula to solve for tan(theta):
tan^2(theta) = sec^2(theta) - 1
Now, substitute x = 2sec(theta):
tan^2(theta) = (2sec(theta))^2 - 1
= 4sec^2(theta) - 1
Since x = 2sec(theta), we can substitute sec^2(theta) with (1 + tan^2(theta)):
tan^2(theta) = 4(1 + tan^2(theta)) - 1
Expanding and rearranging the formula, we get:
tan^2(theta) = 4 + 4tan^2(theta) - 1
3tan^2(theta) = 3
tan^2(theta) = 1
Taking the square root of both sides, we get:
tan(theta) = ±1
So, we can express tan(theta) in terms of x as either tan(theta) = 1 or tan(theta) = -1.