2cos(x)-3tan(x)=3sec(x)
tan= sin/cos
Let C mean cosine, S mean sine(x)
multiply by C
2C^2-3S=3
2(1-S^2)-3S=3
2S^2+3S+1=0
(2S+1)(S+1)=0
so you have the solution.
To solve the equation 2cos(x) - 3tan(x) = 3sec(x), we'll need to use some trigonometric identities and algebraic manipulations.
Let's start by rewriting everything in terms of sine and cosine using the appropriate trigonometric identities:
2cos(x) - 3tan(x) = 3sec(x)
Rewriting tan(x) as sin(x)/cos(x) and sec(x) as 1/cos(x):
2cos(x) - 3(sin(x)/cos(x)) = 3(1/cos(x))
Now, we need to simplify the equation by multiplying through by the common denominator, which is cos(x):
2cos²(x) - 3sin(x) = 3
Next, let's simplify further by using the Pythagorean identity sin²(x) + cos²(x) = 1:
2(1 - sin²(x)) - 3sin(x) = 3
Distributing the 2 to (1 - sin²(x)):
2 - 2sin²(x) - 3sin(x) = 3
Rearranging the terms:
-2sin²(x) - 3sin(x) + 2 - 3 = 0
Combining like terms:
-2sin²(x) - 3sin(x) - 1 = 0
This is now a quadratic equation in terms of sin(x). Let's rewrite it as:
2sin²(x) + 3sin(x) + 1 = 0
Now that we have a quadratic equation, we can solve it by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most convenient method:
(2sin(x) + 1)(sin(x) + 1) = 0
Now, we set each factor equal to zero and solve for sin(x):
1) 2sin(x) + 1 = 0
Subtracting 1 from both sides:
2sin(x) = -1
Dividing both sides by 2:
sin(x) = -1/2
Using the unit circle or trigonometric identities, we can find the solutions for sin(x) = -1/2:
x = -π/6 + 2πn or x = -5π/6 + 2πn, where n is an integer.
2) sin(x) + 1 = 0
Subtracting 1 from both sides:
sin(x) = -1
Using the unit circle or trigonometric identities, we can find the solution for sin(x) = -1:
x = -π/2 + 2πn, where n is an integer.
These are the solutions for the equation 2cos(x) - 3tan(x) = 3sec(x).