To solve the given equation, we can first simplify by writing sin^x in terms of cos^x using the Pythagorean identity: sin^2(x) = 1 - cos^2(x).
The original equation is:2cos^x - sin(x)cos(x) + 5(1 - cos^2(x)) = 3
Expanding and simplifying:2cos^x - sin(x)cos(x) + 5 - 5cos^2(x) = 32cos^x - sin(x)cos(x) - 5cos^2(x) + 5 = 3
Rearranging:5cos^2(x) - 2cos^x + sin(x)cos(x) - 2 = 0
This is a quadratic equation in terms of cos(x). Let cos(x) = u. Then, the equation becomes:5u^2 - 2u + sin(x)u - 2 = 0
Since sin(x) = √(1 - cos^2(x)), we substitute this back into the equation:5u^2 - 2u + √(1 - u^2)u - 2 = 0
This equation can be solved to find the value(s) of cos(x) that satisfy the original equation.
To solve the given equation, we can first simplify by writing sin^x in terms of cos^x using the Pythagorean identity: sin^2(x) = 1 - cos^2(x).
The original equation is:
2cos^x - sin(x)cos(x) + 5(1 - cos^2(x)) = 3
Expanding and simplifying:
2cos^x - sin(x)cos(x) + 5 - 5cos^2(x) = 3
2cos^x - sin(x)cos(x) - 5cos^2(x) + 5 = 3
Rearranging:
5cos^2(x) - 2cos^x + sin(x)cos(x) - 2 = 0
This is a quadratic equation in terms of cos(x). Let cos(x) = u. Then, the equation becomes:
5u^2 - 2u + sin(x)u - 2 = 0
Since sin(x) = √(1 - cos^2(x)), we substitute this back into the equation:
5u^2 - 2u + √(1 - u^2)u - 2 = 0
This equation can be solved to find the value(s) of cos(x) that satisfy the original equation.