Simplify sin(2theta)sec(theta)^4

$\sin (2 θ) \sec^{4} (θ)$

Step 1

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

$\sin (2 θ) {(\frac{1}{\cos (θ)})}^{4}$

Step 2

Simplify the expression.

Apply the product rule to $\frac{1}{\cos (θ)}$ .

$\sin (2 θ) \frac{1^{4}}{\cos^{4} (θ)}$

One to any power is one.

$\sin (2 θ) \frac{1}{\cos^{4} (θ)}$

Step 3

Combine $\sin (2 θ)$ and $\frac{1}{\cos^{4} (θ)}$ .

$\frac{\sin (2 θ)}{\cos^{4} (θ)}$

Step 4

Apply the sine double-angle identity.

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

Step 5

Cancel the common factor of $\cos (θ)$ and $\cos^{4} (θ)$ .

Factor $\cos (θ)$ out of $2 \sin (θ) \cos (θ)$ .

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

Cancel the common factors.

Factor $\cos (θ)$ out of $\cos^{4} (θ)$ .

$\frac{\cos (θ) (2 \sin (θ))}{\cos (θ) \cos^{3} (θ)}$

Cancel the common factor.

$\frac{\cos (θ) (2 \sin (θ))}{\cos (θ) \cos^{3} (θ)}$

Rewrite the expression.

$\frac{2 \sin (θ)}{\cos^{3} (θ)}$

Step 6

Factor $\cos (θ)$ out of $\cos^{3} (θ)$ .

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

Step 7

Separate fractions.

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

Step 8

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

$\frac{2}{\cos^{2} (θ)} \tan (θ)$

Step 9

Multiply by $1$ .

$\frac{2}{\cos^{2} (θ) \cdot 1} \tan (θ)$

Step 10

Separate fractions.

$\frac{2}{1} \cdot \frac{1}{\cos^{2} (θ)} \tan (θ)$

Step 11

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

$\frac{2}{1} \sec^{2} (θ) \tan (θ)$

Step 12

Divide $2$ by $1$ .

$2 \sec^{2} (θ) \tan (θ)$

Simplify sin(2theta)sec(theta)^4

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