How do I solve for x?
cos(4x)(2+tan^2x)=8
To solve for x in the given equation, follow these steps:
Step 1: Simplify the equation.
Use the trigonometric identity: tan^2x = sec^2x - 1.
The equation becomes: cos(4x)(2+(sec^2x - 1)) = 8.
Simplifying further: cos(4x)(sec^2x + 1) = 8.
Step 2: Apply the double-angle identity.
Use the identity: cos(2θ) = 2cos^2(θ) - 1.
The equation can be rewritten as: 2cos^2(2x)cos^2(x) = 8/(sec^2(x) + 1).
Step 3: Simplify and rearrange the equation.
Divide both sides by 2:
cos^2(2x)cos^2(x) = 4/(sec^2(x) + 1).
Step 4: Apply additional identities.
Use the following identities:
cos^2(2x) = (1 + cos(4x))/2, and sec^2(x) = 1 + tan^2(x).
The equation now becomes:
[(1 + cos(4x))/2] * cos^2(x) = 4/[1 + (1 + tan^2(x))] simplified to:
(1 + cos(4x))*cos^2(x) = 8/[1 + (1 + tan^2(x))].
Step 5: Continue simplifying.
Expand the equation: cos^2(x) + cos^3(x)*cos(4x) = 8/[2 + tan^2(x)] + 8.
Step 6: Convert the equation to a quadratic equation.
Rearrange the equation:
cos^3(x)*cos(4x) + cos^2(x) - 8/[2 + tan^2(x)] - 8 = 0.
The equation is now in a form that can be solved either analytically or numerically, depending on the level of simplification you want to achieve.
It's important to note that the resulting equation is quite complex, and an exact analytical solution may be difficult to obtain. Using numerical methods, such as graphing or numerical approximation algorithms, might be more practical to find an approximate solution.