Simplify cos(theta)csc(theta)(sec(theta)^2-1)

$\cos (θ) \cdot \csc (θ) \cdot (\sec^{2} (θ) - 1)$

Step 1

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

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

Step 2

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

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

Step 3

Apply pythagorean identity.

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

Step 4

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

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

Step 5

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

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

Step 6

Combine.

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

Step 7

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

Cancel the common factor of $\sin^{2} (θ)$ and $\sin (θ)$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sin (θ)}{\cos (θ)}$

Step 9

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

$\tan (θ)$

Simplify cos(theta)*csc(theta)*(sec(theta)^2-1)