find x,
tanxsinx= 3sinx-secx
To solve the equation tan(x)sin(x) = 3sin(x) - sec(x) for x, we need to simplify and rearrange the equation.
First, let's convert sec(x) into its equivalent expression in terms of sin(x) and cos(x). Recall that sec(x) = 1/cos(x). Substituting this in the equation, we get:
tan(x)sin(x) = 3sin(x) - 1/cos(x)
Next, let's multiply both sides of the equation by cos(x) to eliminate the denominator:
tan(x)sin(x) * cos(x) = 3sin(x)cos(x) - 1
Using the trigonometric identity tan(x) = sin(x)/cos(x) and simplifying, we get:
sin^2(x) = 3sin(x)cos(x) - 1
Now, let's use the trigonometric identity sin^2(x) + cos^2(x) = 1 to substitute sin^2(x) = 1 - cos^2(x) into the equation:
1 - cos^2(x) = 3sin(x)cos(x) - 1
Rearranging the terms, we have:
cos^2(x) + 3sin(x)cos(x) - 2 = 0
This equation is a quadratic equation in terms of cos(x). Let's solve it by factoring:
(cos(x) + 2)(cos(x) - 1) = 0
Setting each factor equal to zero, we get two possibilities:
cos(x) + 2 = 0 or cos(x) - 1 = 0
Solving these equations, we find:
cos(x) = -2 or cos(x) = 1
However, the range of cosine function is -1 ≤ cos(x) ≤ 1. Since cos(x) = -2 is not within this range, we discard it as an extraneous solution.
Now, let's solve cos(x) = 1. Taking the inverse cosine (arccos) of both sides, we get:
x = arccos(1)
Since the cosine of any angle θ is equal to 1 only when θ = 0 radians, the solution is:
x = 0