To simplify the expression

$\cos x + \cos x + \frac{\sin^2 x}{\cos x}$

let's break down each term and apply trigonometric identities where possible.

Step 1: Combine Like Terms

We see that the expression includes two instances of $\cos x$, so we can combine them:

$\cos x + \cos x = 2 \cos x$

The expression now becomes:

$2 \cos x + \frac{\sin^2 x}{\cos x}$

Step 2: Separate the Terms

Rewrite this as a single fraction to see if it can be simplified further:

$\frac{2 \cos^2 x + \sin^2 x}{\cos x}$

Step 3: Apply the Pythagorean Identity

We know that $\sin^2 x + \cos^2 x = 1$. Rewrite $\sin^2 x$ using this identity:

$= \frac{2 \cos^2 x + (1 - \cos^2 x)}{\cos x}$

Step 4: Simplify the Numerator

Combine terms in the numerator:

$= \frac{2 \cos^2 x + 1 - \cos^2 x}{\cos x} = \frac{\cos^2 x + 1}{\cos x}$

Step 5: Split the Fraction

Now, separate the terms:

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

Final Simplified Expression

The expression simplifies to:

$\cos x + \sec x$

Matching Options

None of the options exactly match $\cos x + \sec x$, so we might check if this form corresponds to another option in an alternate interpretation.

Would you like further verification or have questions on another approach?

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Questions for Further Practice

1. Simplify $\sin^2 x + \cos^2 x$ using identities.
2. Verify the identity $\sec^2 x - \tan^2 x = 1$.
3. Simplify $\tan x \sec x$ using trigonometric identities.
4. Expand and simplify $(\sin x + \cos x)^2$.
5. Rewrite $\sec x \tan x$ in terms of $\sin x$ and $\cos x$.

Tip

Remember that identities like $\sin^2 x + \cos^2 x = 1$ are helpful for rewriting expressions in simpler forms.