Simplify cos(x)(1+tan(2)(x))

$\cos (x) (1 + \tan (2) (x))$

Step 1

Simplify each term.

Rewrite $\tan (2)$ in terms of sines and cosines.

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

Combine $\frac{\sin (2)}{\cos (2)}$ and $x$ .

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

Step 2

Simplify terms.

Apply the distributive property.

$\cos (x) \cdot 1 + \cos (x) \frac{\sin (2) x}{\cos (2)}$

Multiply $\cos (x)$ by $1$ .

$\cos (x) + \cos (x) \frac{\sin (2) x}{\cos (2)}$

Combine $\cos (x)$ and $\frac{\sin (2) x}{\cos (2)}$ .

$\cos (x) + \frac{\cos (x) (\sin (2) x)}{\cos (2)}$

Step 3

Simplify each term.

Separate fractions.

$\cos (x) + \frac{\sin (2) x}{1} \cdot \frac{\cos (x)}{\cos (2)}$

Rewrite $\frac{\cos (x)}{\cos (2)}$ as a product.

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

Write $\cos (x)$ as a fraction with denominator $1$ .

$\cos (x) + \frac{\sin (2) x}{1} (\frac{\cos (x)}{1} \cdot \frac{1}{\cos (2)})$

Simplify.

Divide $\cos (x)$ by $1$ .

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

Convert from $\frac{1}{\cos (2)}$ to $\sec (2)$ .

$\cos (x) + \frac{\sin (2) x}{1} (\cos (x) \sec (2))$

Rewrite using the commutative property of multiplication.

$\cos (x) + \sec (2) \frac{\sin (2) x}{1} \cos (x)$

Divide $\sin (2) x$ by $1$ .

$\cos (x) + \sec (2) (\sin (2) x) \cos (x)$

Rewrite $\sec (2)$ in terms of sines and cosines.

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

Combine $\frac{1}{\cos (2)}$ and $\sin (2)$ .

$\cos (x) + \frac{\sin (2)}{\cos (2)} x \cos (x)$

Convert from $\frac{\sin (2)}{\cos (2)}$ to $\tan (2)$ .

$\cos (x) + \tan (2) x \cos (x)$

Step 4

Reorder factors in $\cos (x) + \tan (2) x \cos (x)$ .

$\cos (x) + x \tan (2) \cos (x)$