(sin x + cos x)(tan² x + 1)/tan x = 1/cos x + 1/sin x

Steps to Solve

1. Rewrite the left side

Start with the left side of the equation:

$$(\sin x + \cos x) \left( \frac{\tan^2 x + 1}{\tan x} \right)$$

We know that $\tan^2 x + 1 = \sec^2 x$, so we can rewrite it as:

$$(\sin x + \cos x) \left( \frac{\sec^2 x}{\tan x} \right)$$

2. Substitute the definitions of sine, cosine, and tangent

Recall the definitions of the trigonometric functions:

$$\tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \sec x = \frac{1}{\cos x}$$

Thus:

$$\sec^2 x = \frac{1}{\cos^2 x}$$

Substituting these into our expression, we get:

$$(\sin x + \cos x) \left( \frac{\frac{1}{\cos^2 x}}{\frac{\sin x}{\cos x}} \right)$$

3. Simplify the expression

This simplifies to:

$$(\sin x + \cos x) \left( \frac{1}{\cos^2 x} \cdot \frac{\cos x}{\sin x} \right) = \frac{\sin x + \cos x}{\cos x \sin x}$$

4. Rewrite the right-hand side

Now, consider the right-hand side of the original equation:

$$ \frac{1}{\cos x} + \frac{1}{\sin x} $$

Finding a common denominator:

$$ \frac{\sin x + \cos x}{\sin x \cos x} $$

5. Set both sides equal

Now we can see that both sides are equal:

$$ \frac{\sin x + \cos x}{\sin x \cos x} = \frac{\sin x + \cos x}{\sin x \cos x} $$

This confirms the equation holds true.