Algebra Examples

$1 + 2 \cot^{2} (θ) + \cot^{4} (θ) = {\csc (θ)}^{n}$

Step 1

Rewrite the equation as ${\csc (θ)}^{n} = 1 + 2 \cot^{2} (θ) + \cot^{4} (θ)$ .

${\csc (θ)}^{n} = 1 + 2 \cot^{2} (θ) + \cot^{4} (θ)$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({\csc (θ)}^{n}) = \ln (1 + 2 \cot^{2} (θ) + \cot^{4} (θ))$

Step 3

Expand $\ln ({\csc (θ)}^{n})$ by moving $n$ outside the logarithm.

$n \ln (\csc (θ)) = \ln (1 + 2 \cot^{2} (θ) + \cot^{4} (θ))$

Step 4

Divide each term in $n \ln (\csc (θ)) = \ln (1 + 2 \cot^{2} (θ) + \cot^{4} (θ))$ by $\ln (\csc (θ))$ and simplify.

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Step 4.1

Divide each term in $n \ln (\csc (θ)) = \ln (1 + 2 \cot^{2} (θ) + \cot^{4} (θ))$ by $\ln (\csc (θ))$ .

$\frac{n \ln (\csc (θ))}{\ln (\csc (θ))} = \frac{\ln (1 + 2 \cot^{2} (θ) + \cot^{4} (θ))}{\ln (\csc (θ))}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $\ln (\csc (θ))$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{n \ln (\csc (θ))}{\ln (\csc (θ))} = \frac{\ln (1 + 2 \cot^{2} (θ) + \cot^{4} (θ))}{\ln (\csc (θ))}$

Step 4.2.1.2

Divide $n$ by $1$ .

$n = \frac{\ln (1 + 2 \cot^{2} (θ) + \cot^{4} (θ))}{\ln (\csc (θ))}$