1. **State the problem:** Prove the trigonometric identity $$\sin x \tan x + \cos x = \sec x$$. 2. **Recall the definitions and formulas:** - $$\tan x = \frac{\sin x}{\cos x}$$ - $$\sec x = \frac{1}{\cos x}$$ 3. **Substitute $$\tan x$$ in the left-hand side (LHS):** $$\sin x \tan x + \cos x = \sin x \cdot \frac{\sin x}{\cos x} + \cos x$$ 4. **Simplify the expression:** $$= \frac{\sin^2 x}{\cos x} + \cos x$$ 5. **Write $$\cos x$$ as $$\frac{\cos^2 x}{\cos x}$$ to have a common denominator:** $$= \frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\cos x}$$ 6. **Combine the fractions:** $$= \frac{\sin^2 x + \cos^2 x}{\cos x}$$ 7. **Use the Pythagorean identity $$\sin^2 x + \cos^2 x = 1$$:** $$= \frac{1}{\cos x}$$ 8. **Recognize that $$\frac{1}{\cos x} = \sec x$$, which is the right-hand side (RHS).** **Therefore, the identity is proven:** $$\sin x \tan x + \cos x = \sec x$$