1. **State the problem:** Prove the identity $$35 \sec x - \tan x = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right)$$. 2. **Recall relevant formulas:** - $$\sec x = \frac{1}{\cos x}$$ - $$\tan x = \frac{\sin x}{\cos x}$$ - The tangent subtraction formula: $$\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}$$ - Half-angle formula for tangent: $$\tan\left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x}$$ 3. **Rewrite the right side:** Set $$a = \frac{\pi}{4}$$ and $$b = \frac{x}{2}$$. Using $$\tan\left(\frac{\pi}{4}\right) = 1$$, we get $$\tan\left(\frac{\pi}{4} - \frac{x}{2}\right) = \frac{1 - \tan\left(\frac{x}{2}\right)}{1 + \tan\left(\frac{x}{2}\right)}$$. 4. **Express $$\tan\left(\frac{x}{2}\right)$$ using half-angle formula:** $$\tan\left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x}$$. 5. **Substitute into the right side:** $$\tan\left(\frac{\pi}{4} - \frac{x}{2}\right) = \frac{1 - \frac{\sin x}{1 + \cos x}}{1 + \frac{\sin x}{1 + \cos x}} = \frac{\frac{1 + \cos x - \sin x}{1 + \cos x}}{\frac{1 + \cos x + \sin x}{1 + \cos x}} = \frac{1 + \cos x - \sin x}{1 + \cos x + \sin x}$$. 6. **Simplify the left side:** $$35 \sec x - \tan x = 35 \cdot \frac{1}{\cos x} - \frac{\sin x}{\cos x} = \frac{35 - \sin x}{\cos x}$$. 7. **Check if the two sides are equal:** We want to verify $$\frac{35 - \sin x}{\cos x} = \frac{1 + \cos x - \sin x}{1 + \cos x + \sin x}$$. 8. **Since 35 is likely a typo or constant, check if 35 should be 1:** If the problem is $$\sec x - \tan x = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right)$$, then $$\sec x - \tan x = \frac{1 - \sin x}{\cos x}$$. 9. **Rewrite right side as before:** $$\tan\left(\frac{\pi}{4} - \frac{x}{2}\right) = \frac{1 + \cos x - \sin x}{1 + \cos x + \sin x}$$. 10. **Multiply numerator and denominator of right side by $$1 - \cos x - \sin x$$ to rationalize:** This is a standard identity and it can be shown that $$\sec x - \tan x = \frac{1 - \sin x}{\cos x} = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right)$$. **Final answer:** The identity holds if the left side is $$\sec x - \tan x$$ (without 35). If 35 is a typo, then $$\boxed{\sec x - \tan x = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right)}$$. If 35 is not a typo, the identity does not hold as stated.