How do I find 2 tan squared time x = sec times x -1
To solve the equation 2tan^2(x) = sec(x) - 1, let's break down the steps:
Step 1: Start by using the fundamentals of trigonometric identities and equations. Recall that tan(x) can be expressed as sin(x) / cos(x) and sec(x) can be expressed as 1 / cos(x).
Step 2: Rewrite the equation in terms of sin(x) and cos(x):
2(sin^2(x) / cos^2(x)) = 1 / cos(x) - 1
Step 3: Simplify the equation by multiplying both sides by cos^2(x) to eliminate the denominators:
2sin^2(x) = 1 - cos(x)
Step 4: Use the identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) in the equation:
2(1 - cos^2(x)) = 1 - cos(x)
Step 5: Distribute and simplify:
2 - 2cos^2(x) = 1 - cos(x)
Step 6: Rearrange the equation to bring all terms to one side:
2cos^2(x) - cos(x) - 1 = 0
Step 7: Now, this is a quadratic equation in terms of cos(x). We can solve it by factoring, completing the square, or using the quadratic formula.
In this case, the quadratic equation is factorable:
(2cos(x) + 1)(cos(x) - 1) = 0
Step 8: Set each factor to zero and solve for cos(x):
2cos(x) + 1 = 0 --> cos(x) = -1/2
cos(x) - 1 = 0 --> cos(x) = 1
Step 9: Find the solutions for x by using the inverse cosine function (also known as arccosine):
cos(x) = -1/2 --> x = arccos(-1/2)
cos(x) = 1 --> x = arccos(1)
Step 10: Lastly, find all possible solutions for x by considering the domain of the inverse cosine function, which is restricted to x ∈ [0, π].
Therefore, x can take the values:
x = arccos(-1/2) (if it falls within the valid domain)
x = arccos(1)
Note: Remember to convert the inverse cosine values to degrees or radians, depending on the context or specific instructions given.