Solve for ? tan(x)+cot(x)=csc(x)sec(x)

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Trigonometry

$\tan (x) + \cot (x) = \csc (x) \sec (x)$

Since $\sec (x)$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\csc (x) \sec (x) = \tan (x) + \cot (x)$

Divide each term by $\csc (x)$ and simplify.

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Divide each term in $\csc (x) \sec (x) = \tan (x) + \cot (x)$ by $\csc (x)$ .

$\frac{\csc (x) \sec (x)}{\csc (x)} = \frac{\tan (x)}{\csc (x)} + \frac{\cot (x)}{\csc (x)}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{\csc (x) \sec (x)}{\csc (x)} = \frac{\tan (x)}{\csc (x)} + \frac{\cot (x)}{\csc (x)}$

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

$\sec (x) = \frac{\tan (x)}{\csc (x)} + \frac{\cot (x)}{\csc (x)}$

Simplify each term.

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

$\sec (x) = \frac{\tan (x)}{\frac{1}{\sin (x)}} + \frac{\cot (x)}{\csc (x)}$

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

$\sec (x) = \frac{\frac{\sin (x)}{\cos (x)}}{\frac{1}{\sin (x)}} + \frac{\cot (x)}{\csc (x)}$

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\sec (x) = \frac{\sin (x)}{\cos (x)} \cdot \sin (x) + \frac{\cot (x)}{\csc (x)}$

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

$\sec (x) = \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{1} + \frac{\cot (x)}{\csc (x)}$

Simplify.

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

$\sec (x) = \frac{\sin (x)}{\cos (x)} \cdot \sin (x) + \frac{\cot (x)}{\csc (x)}$

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

$\sec (x) = \tan (x) \sin (x) + \frac{\cot (x)}{\csc (x)}$

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

$\sec (x) = \tan (x) \sin (x) + \frac{\cot (x)}{\frac{1}{\sin (x)}}$

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

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

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

$\sec (x) = \tan (x) \sin (x) + \frac{\cos (x)}{\sin (x)} \cdot \sin (x)$

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

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

Cancel the common factor of $\sin (x)$ .

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Factor out the greatest common factor $\sin (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

Divide $1$ by $1$ .

$\sec (x) = \tan (x) \sin (x) + \cos (x) \cdot 1$

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

$\sec (x) = \tan (x) \sin (x) + \cos (x)$

Simplify $\tan (x) \sin (x) + \cos (x)$ .

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Simplify each term.

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

$\sec (x) = \frac{\sin (x)}{\cos (x)} \cdot \sin (x) + \cos (x)$

Simplify $\frac{\sin (x)}{\cos (x)} \sin (x)$ .

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

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

Multiply $\frac{\sin (x)}{\cos (x)}$ and $\frac{\sin (x)}{1}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

Separate fractions.

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

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

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

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

$\sec (x) = \sin (x) \tan (x) + \cos (x)$

Divide each term in the equation by $\cos (x)$ .

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

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

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

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

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

Multiply $\frac{1}{\cos (x)}$ and $\frac{1}{\cos (x)}$ .

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

Simplify the denominator.

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

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

Add $1$ and $1$ .

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

Simplify with factoring out.

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Rewrite $1$ as $1^{2}$ .

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

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

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

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

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

Replace $\cos (x)$ with an equivalent expression in the numerator.

$\sec^{2} (x) = (\sin (x) \tan (x) + \cos (x)) \cdot \sec (x)$

Remove parentheses.

$\sec^{2} (x) = (\sin (x) \tan (x) + \cos (x)) \cdot \sec (x)$

Multiply $\sin (x) \tan (x) + \cos (x)$ by $\sec (x)$ .

$\sec^{2} (x) = (\sin (x) \tan (x) + \cos (x)) \sec (x)$

Apply the distributive property.

$\sec^{2} (x) = \sin (x) \tan (x) \sec (x) + \cos (x) \sec (x)$

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

$\sec^{2} (x) = \sin (x) (\frac{\sin (x)}{\cos (x)}) \cdot \sec (x) + \cos (x) \sec (x)$

Simplify $\sin (x) \frac{\sin (x)}{\cos (x)}$ .

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

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

Multiply $\frac{\sin (x)}{1}$ and $\frac{\sin (x)}{\cos (x)}$ .

$\sec^{2} (x) = \frac{\sin (x) \sin (x)}{\cos (x)} \cdot \sec (x) + \cos (x) \sec (x)$

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

$\sec^{2} (x) = \frac{\sin (x) \sin (x)}{\cos (x)} \cdot \sec (x) + \cos (x) \sec (x)$

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

$\sec^{2} (x) = \frac{\sin (x) \sin (x)}{\cos (x)} \cdot \sec (x) + \cos (x) \sec (x)$

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

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

Add $1$ and $1$ .

$\sec^{2} (x) = \frac{\sin^{2} (x)}{\cos (x)} \cdot \sec (x) + \cos (x) \sec (x)$

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

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

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

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Multiply $\frac{\sin^{2} (x)}{\cos (x)}$ and $\frac{1}{\cos (x)}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

$\sec^{2} (x) = \tan^{2} (x) + \cos (x) \sec (x)$

Rewrite in terms of sines and cosines, then cancel the common factors.

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Reorder $\cos (x)$ and $\sec (x)$ .

$\sec^{2} (x) = \tan^{2} (x) + \sec (x) \cos (x)$

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

$\sec^{2} (x) = \tan^{2} (x) + \frac{1}{\cos (x)} \cdot \cos (x)$

Cancel the common factors.

$\sec^{2} (x) = \tan^{2} (x) + 1$

Replace $1 + t a n^{2} (x)$ with $s e c^{2} (x)$ using the identity $s e c^{2} (x) = 1 + t a n^{2} (x)$ .

$\sec^{2} (x) = \sec^{2} (x)$

This is the proper way of writing a trigonometric expression raised to an exponent.

$\sec^{2} (x) = 1 \sec^{2} (x)$

Multiply $\sec^{2} (x)$ by $1$ .

$\sec^{2} (x) = \sec^{2} (x)$

This is the proper way of writing a trigonometric expression raised to an exponent.

$\sec^{2} (x) = 1 \sec^{2} (x)$

Multiply $\sec^{2} (x)$ by $1$ .

$\sec^{2} (x) = \sec^{2} (x)$

This is the proper way of writing a trigonometric expression raised to an exponent.

$\sec^{2} (x) = 1 \sec^{2} (x)$

Multiply $\sec^{2} (x)$ by $1$ .

$\sec^{2} (x) = \sec^{2} (x)$

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Move all terms containing $x$ to the left side of the equation.

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Subtract $x$ from both sides of the equation.

$x - x = 0$

Subtract $x$ from $x$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

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