\( \tan x ( 1 - \cos ^ { 2 } x ) + \cos ^ { 2 } x = \frac { ( \sin x + \frac { \cos x ) ( 1 - \sin x \cos x } { \cos x } } { 3 } \)

The equation provided is quite complex, and attempts to solve it directly have resulted in timeouts. Let's break it down step by step to simplify and analyze it. ### Step 1: Rewrite the Equation The original equation is: \[ \tan x (1 - \cos^2 x) + \cos^2 x = \frac{\sin x + \cos x (1 - \sin x \cos x)}{3} \] ### Step 2: Use Trigonometric Identities Recall that: \[ \tan x = \frac{\sin x}{\cos x} \] \[ 1 - \cos^2 x = \sin^2 x \] Substituting these into the equation gives: \[ \frac{\sin x}{\cos x} \sin^2 x + \cos^2 x = \frac{\sin x + \cos x (1 - \sin x \cos x)}{3} \] ### Step 3: Simplify the Left Side The left side becomes: \[ \frac{\sin^3 x}{\cos x} + \cos^2 x \] ### Step 4: Simplify the Right Side The right side can be simplified as follows: \[ \frac{\sin x + \cos x - \sin x \cos^2 x}{3} \] ### Step 5: Combine and Rearrange Now we have: \[ \frac{\sin^3 x}{\cos x} + \cos^2 x = \frac{\sin x + \cos x - \sin x \cos^2 x}{3} \] To eliminate the fraction, multiply through by \(3\cos x\): \[ 3\sin^3 x + 3\cos^3 x = \sin x \cos x + 3\cos^2 x - \sin x \cos^3 x \] ### Step 6: Rearranging the Equation Rearranging gives us: \[ 3\sin^3 x + 3\cos^3 x - \sin x \cos x - 3\cos^2 x + \sin x \cos^3 x = 0 \] ### Step 7: Solve the Simplified Equation This equation is still complex, but we can try to find specific values of \(x\) that satisfy it, such as \(x = 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi\), etc. Let's evaluate the equation at these points to see if any of them are solutions. 1. **At \(x = 0\)**: \[ 3(0)^3 + 3(1)^3 - 0 - 3(1)^2 + 0 = 0 \Rightarrow 3 - 3 = 0 \quad \text{(True)} \] 2. **At \(x = \frac{\pi}{4}\)**: \[ 3\left(\frac{\sqrt{2}}{2}\right)^3 + 3\left(\frac{\sqrt{2}}{2}\right)^3 - \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2}}{2} - 3\left(\frac{\sqrt{2}}{2}\right)^2 + \frac{\sqrt{2}}{2}\left(\frac{\sqrt{2}}{2}\right)^3 = 0 \] This will require calculation. 3. **At \(x = \frac{\pi}{2}\)**: \[ 3(1)^3 + 3(0)^3 - 0 - 3(0)^2 + 0 = 0 \Rightarrow 3 \neq 0 \quad \text{(False)} \] 4. **At \(x = \pi\)**: \[ 3(0)^3 + 3(-1)^3 - 0 - 3(1)^2 + 0 = 0 \Rightarrow -3 - 3 \neq 0 \quad \text{(False)} \] ### Conclusion From the evaluations, \(x = 0\) is a solution. Further numerical or graphical methods may be needed to find additional solutions or confirm the behavior of the function. If you would like to explore specific values or methods further, please let me know!