Answer: To show that the equation "3sin () tan () = 5cos () - 2" is equivalent to the equation "(4 cos() - 3)(2 cos () + 1) = 0", we need to simplify the first equation and check if it has the same solutions as the second equation.
Starting with the first equation:
3sin () tan () = 5cos () - 2
Using the identity tan () = sin () / cos (), we can write:
3sin () (sin () / cos ()) = 5cos () - 2
Multiplying both sides by cos (), we get:
3sin^2 () = (5cos () - 2)cos ()
Using the identity sin^2 () + cos^2 () = 1 and rearranging, we get:
3(1 - cos^2 ()) = 5cos^2 () - 2cos ()
Expanding and rearranging, we get:
5cos^2 () - 2cos () - 3 + 3cos^2 () = 0
Simplifying, we get:
8cos^2 () - 2cos () - 3 = 0
Now, we can use the quadratic formula to solve for cos ():
cos () = [2 ± sqrt(2^2 - 4(8)(-3))]/(2(8))
cos () = [2 ± sqrt(100)]/16
cos () = (1/4) or (-3/8)
Substituting these values back into the original equation, we can verify that they satisfy the equation.
Now, let's consider the second equation:
(4 cos() - 3)(2 cos () + 1) = 0
This equation is satisfied when either 4cos() - 3 = 0 or 2cos() + 1 = 0.
Solving for cos() in the first equation, we get:
4cos() - 3 = 0
cos() = 3/4
Substituting this value back into the original equation, we can verify that it satisfies the equation.
Solving for cos() in the second equation, we get:
2cos() + 1 = 0
cos() = -1/2
Substituting this value back into the original equation, we can also verify that it satisfies the equation.
Therefore, we have shown that the equation "3sin () tan () = 5cos () - 2" is equivalent to the equation "(4 cos() - 3)(2 cos () + 1) = 0".