Simplify (tan(theta)+cos(theta)csc(theta))/(sec(theta)csc(theta))

$\frac{\tan (θ) + \cos (θ) \csc (θ)}{\sec (θ) \csc (θ)}$

Step 1

Simplify the numerator.

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

$\frac{\frac{\sin (θ)}{\cos (θ)} + \cos (θ) \csc (θ)}{\sec (θ) \csc (θ)}$

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

$\frac{\frac{\sin (θ)}{\cos (θ)} + \cos (θ) \frac{1}{\sin (θ)}}{\sec (θ) \csc (θ)}$

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

$\frac{\frac{\sin (θ)}{\cos (θ)} + \frac{\cos (θ)}{\sin (θ)}}{\sec (θ) \csc (θ)}$

Step 2

Simplify the denominator.

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

$\frac{\frac{\sin (θ)}{\cos (θ)} + \frac{\cos (θ)}{\sin (θ)}}{\frac{1}{\cos (θ)} \csc (θ)}$

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

$\frac{\frac{\sin (θ)}{\cos (θ)} + \frac{\cos (θ)}{\sin (θ)}}{\frac{1}{\cos (θ)} \cdot \frac{1}{\sin (θ)}}$

Step 3

Multiply $\frac{1}{\cos (θ)}$ by $\frac{1}{\sin (θ)}$ .

$\frac{\frac{\sin (θ)}{\cos (θ)} + \frac{\cos (θ)}{\sin (θ)}}{\frac{1}{\cos (θ) \sin (θ)}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Apply the distributive property.

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

Step 6

Cancel the common factor of $\cos (θ)$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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

Step 11

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

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

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

Cancel the common factor.

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

Rewrite the expression.

$\sin^{2} (θ) + \cos (θ) \cos (θ)$

Step 12

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

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

Step 13

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

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

Step 14

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

$\sin^{2} (θ) + {\cos (θ)}^{1 + 1}$

Step 15

Add $1$ and $1$ .

$\sin^{2} (θ) + \cos^{2} (θ)$

Step 16

Apply pythagorean identity.

$1$

Simplify (tan(theta)+cos(theta)csc(theta))/(sec(theta)csc(theta))

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