Find the Exact Value tan(60)sec(30)+sin(45)^2tan(30)^2

$\tan (60) \sec (30) + \sin^{2} (45) \tan^{2} (30)$

Simplify each term.

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\sqrt{3} \sec (30) + \sin^{2} (45) \tan^{2} (30)$

The exact value of $\sec (30)$ is $\frac{2}{\sqrt{3}}$ .

$\sqrt{3} \frac{2}{\sqrt{3}} + \sin^{2} (45) \tan^{2} (30)$

Cancel the common factor of $\sqrt{3}$ .

Cancel the common factor.

$\sqrt{3} \frac{2}{\sqrt{3}} + \sin^{2} (45) \tan^{2} (30)$

Rewrite the expression.

$2 + \sin^{2} (45) \tan^{2} (30)$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$2 + {(\frac{\sqrt{2}}{2})}^{2} \tan^{2} (30)$

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$2 + \frac{{\sqrt{2}}^{2}}{2^{2}} \tan^{2} (30)$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$2 + \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} \tan^{2} (30)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 + \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} \tan^{2} (30)$

Combine $\frac{1}{2}$ and $2$ .

$2 + \frac{2^{\frac{2}{2}}}{2^{2}} \tan^{2} (30)$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 + \frac{2^{\frac{2}{2}}}{2^{2}} \tan^{2} (30)$

Rewrite the expression.

$2 + \frac{2^{1}}{2^{2}} \tan^{2} (30)$

Evaluate the exponent.

$2 + \frac{2}{2^{2}} \tan^{2} (30)$

Raise $2$ to the power of $2$ .

$2 + \frac{2}{4} \tan^{2} (30)$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$2 + \frac{2 (1)}{4} \tan^{2} (30)$

Cancel the common factors.

Factor $2$ out of $4$ .

$2 + \frac{2 \cdot 1}{2 \cdot 2} \tan^{2} (30)$

Cancel the common factor.

$2 + \frac{2 \cdot 1}{2 \cdot 2} \tan^{2} (30)$

Rewrite the expression.

$2 + \frac{1}{2} \tan^{2} (30)$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$2 + \frac{1}{2} {(\frac{\sqrt{3}}{3})}^{2}$

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

$2 + \frac{1}{2} \cdot \frac{{\sqrt{3}}^{2}}{3^{2}}$

Combine.

$2 + \frac{1 {\sqrt{3}}^{2}}{2 \cdot 3^{2}}$

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$2 + \frac{{\sqrt{3}}^{2}}{2 \cdot 3^{2}}$

Raise $3$ to the power of $2$ .

$2 + \frac{{\sqrt{3}}^{2}}{2 \cdot 9}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$2 + \frac{{(3^{\frac{1}{2}})}^{2}}{2 \cdot 9}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 + \frac{3^{\frac{1}{2} \cdot 2}}{2 \cdot 9}$

Combine $\frac{1}{2}$ and $2$ .

$2 + \frac{3^{\frac{2}{2}}}{2 \cdot 9}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 + \frac{3^{\frac{2}{2}}}{2 \cdot 9}$

Rewrite the expression.

$2 + \frac{3^{1}}{2 \cdot 9}$

Evaluate the exponent.

$2 + \frac{3}{2 \cdot 9}$

Multiply $2$ by $9$ .

$2 + \frac{3}{18}$

Cancel the common factor of $3$ and $18$ .

Factor $3$ out of $3$ .

$2 + \frac{3 (1)}{18}$

Cancel the common factors.

Factor $3$ out of $18$ .

$2 + \frac{3 \cdot 1}{3 \cdot 6}$

Cancel the common factor.

$2 + \frac{3 \cdot 1}{3 \cdot 6}$

Rewrite the expression.

$2 + \frac{1}{6}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 \cdot \frac{6}{6} + \frac{1}{6}$

Combine $2$ and $\frac{6}{6}$ .

$\frac{2 \cdot 6}{6} + \frac{1}{6}$

Combine the numerators over the common denominator.

$\frac{2 \cdot 6 + 1}{6}$

Simplify the numerator.

Multiply $2$ by $6$ .

$\frac{12 + 1}{6}$

Add $12$ and $1$ .

$\frac{13}{6}$

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{6}$

Decimal Form:

$2.1\overline{6}$

Mixed Number Form:

$2 \frac{1}{6}$

Find the Exact Value tan(60)sec(30)+sin(45)^2tan(30)^2

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