Simplify tan(x)^2-(sin(x)^2)/(cot(x)^2)

$\tan^{2} (x) - \frac{\sin^{2} (x)}{\cot^{2} (x)}$

Simplify each term.

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

${(\frac{\sin (x)}{\cos (x)})}^{2} - \frac{\sin^{2} (x)}{\cot^{2} (x)}$

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cot^{2} (x)}$

Rewrite $\frac{\sin^{2} (x)}{\cot^{2} (x)}$ as ${(\frac{\sin (x)}{\cot (x)})}^{2}$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - {(\frac{\sin (x)}{\cot (x)})}^{2}$

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - {(\frac{\sin (x)}{\frac{\cos (x)}{\sin (x)}})}^{2}$

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - {(\sin (x) \frac{\sin (x)}{\cos (x)})}^{2}$

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

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

Simplify.

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - {(\sin (x) \frac{\sin (x)}{\cos (x)})}^{2}$

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - {(\frac{\sin (x) \sin (x)}{\cos (x)})}^{2}$

Simplify the numerator.

Raise $\sin (x)$ to the power of $1$ .

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

Raise $\sin (x)$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - {(\frac{\sin^{2} (x)}{\cos (x)})}^{2}$

Apply the product rule to $\frac{\sin^{2} (x)}{\cos (x)}$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{{(\sin^{2} (x))}^{2}}{\cos^{2} (x)}$

Multiply the exponents in ${(\sin^{2} (x))}^{2}$ .

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{{\sin (x)}^{2 \cdot 2}}{\cos^{2} (x)}$

Multiply $2$ by $2$ .

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin^{4} (x)}{\cos^{2} (x)}$

Rewrite $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ as ${(\frac{\sin (x)}{\cos (x)})}^{2}$ .

${(\frac{\sin (x)}{\cos (x)})}^{2} - \frac{\sin^{4} (x)}{\cos^{2} (x)}$

Rewrite $\frac{\sin^{4} (x)}{\cos^{2} (x)}$ as ${(\frac{\sin^{2} (x)}{\cos (x)})}^{2}$ .

${(\frac{\sin (x)}{\cos (x)})}^{2} - {(\frac{\sin^{2} (x)}{\cos (x)})}^{2}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{\sin (x)}{\cos (x)}$ and $b = \frac{\sin^{2} (x)}{\cos (x)}$ .

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

Simplify terms.

Simplify each term.

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

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

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Separate fractions.

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

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

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

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

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

Simplify each term.

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

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

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$(\tan (x) + \sin (x) \tan (x)) (\tan (x) - \frac{\sin (x) \sin (x)}{\cos (x)})$

Separate fractions.

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

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

$(\tan (x) + \sin (x) \tan (x)) (\tan (x) - (\frac{\sin (x)}{1} \tan (x)))$

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

$(\tan (x) + \sin (x) \tan (x)) (\tan (x) - \sin (x) \tan (x))$

Expand $(\tan (x) + \sin (x) \tan (x)) (\tan (x) - \sin (x) \tan (x))$ using the FOIL Method.

Apply the distributive property.

$\tan (x) (\tan (x) - \sin (x) \tan (x)) + \sin (x) \tan (x) (\tan (x) - \sin (x) \tan (x))$

Apply the distributive property.

$\tan (x) \tan (x) + \tan (x) (- \sin (x) \tan (x)) + \sin (x) \tan (x) (\tan (x) - \sin (x) \tan (x))$

Apply the distributive property.

$\tan (x) \tan (x) + \tan (x) (- \sin (x) \tan (x)) + \sin (x) \tan (x) \tan (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Simplify terms.

Combine the opposite terms in $\tan (x) \tan (x) + \tan (x) (- \sin (x) \tan (x)) + \sin (x) \tan (x) \tan (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$ .

Reorder the factors in the terms $\tan (x) (- \sin (x) \tan (x))$ and $\sin (x) \tan (x) \tan (x)$ .

$\tan (x) \tan (x) - \sin (x) \tan (x) \tan (x) + \sin (x) \tan (x) \tan (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Add $- \sin (x) \tan (x) \tan (x)$ and $\sin (x) \tan (x) \tan (x)$ .

$\tan (x) \tan (x) + 0 + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Add $\tan (x) \tan (x)$ and $0$ .

$\tan (x) \tan (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Simplify each term.

Multiply $\tan (x) \tan (x)$ .

Raise $\tan (x)$ to the power of $1$ .

$\tan^{1} (x) \tan (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Raise $\tan (x)$ to the power of $1$ .

$\tan^{1} (x) \tan^{1} (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\tan (x)}^{1 + 1} + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Add $1$ and $1$ .

$\tan^{2} (x) + \sin (x) \tan (x) (- \sin (x) \tan (x))$

Rewrite using the commutative property of multiplication.

$\tan^{2} (x) - \sin (x) \tan (x) \sin (x) \tan (x)$

Multiply $- \sin (x) \tan (x) \sin (x)$ .

Raise $\sin (x)$ to the power of $1$ .

$\tan^{2} (x) - (\sin^{1} (x) \sin (x)) \tan (x) \tan (x)$

Raise $\sin (x)$ to the power of $1$ .

$\tan^{2} (x) - (\sin^{1} (x) \sin^{1} (x)) \tan (x) \tan (x)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan^{2} (x) - {\sin (x)}^{1 + 1} \tan (x) \tan (x)$

Add $1$ and $1$ .

$\tan^{2} (x) - \sin^{2} (x) \tan (x) \tan (x)$

Multiply $- \sin^{2} (x) \tan (x) \tan (x)$ .

Raise $\tan (x)$ to the power of $1$ .

$\tan^{2} (x) - \sin^{2} (x) (\tan^{1} (x) \tan (x))$

Raise $\tan (x)$ to the power of $1$ .

$\tan^{2} (x) - \sin^{2} (x) (\tan^{1} (x) \tan^{1} (x))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan^{2} (x) - \sin^{2} (x) {\tan (x)}^{1 + 1}$

Add $1$ and $1$ .

$\tan^{2} (x) - \sin^{2} (x) \tan^{2} (x)$

Factor $\tan^{2} (x)$ out of $\tan^{2} (x) - \sin^{2} (x) \tan^{2} (x)$ .

Multiply by $1$ .

$\tan^{2} (x) \cdot 1 - \sin^{2} (x) \tan^{2} (x)$

Factor $\tan^{2} (x)$ out of $- \sin^{2} (x) \tan^{2} (x)$ .

$\tan^{2} (x) \cdot 1 + \tan^{2} (x) (- \sin^{2} (x))$

Factor $\tan^{2} (x)$ out of $\tan^{2} (x) \cdot 1 + \tan^{2} (x) (- \sin^{2} (x))$ .

$\tan^{2} (x) (1 - \sin^{2} (x))$

Apply pythagorean identity.

$\tan^{2} (x) \cos^{2} (x)$

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

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

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Cancel the common factor of $\cos^{2} (x)$ .

Cancel the common factor.

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

Rewrite the expression.

$\sin^{2} (x)$

Simplify tan(x)^2-(sin(x)^2)/(cot(x)^2)

Related Questions

Support LCMGCF.com by disabling your adblocker.