Trigonometry Examples

$\frac{1 + \cos (a) \tan (a) \csc (a)}{\sec (a)}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Reorder $\cos (a)$ and $\tan (a)$ .

$\frac{1 + \tan (a) \cos (a) \csc (a)}{\sec (a)}$

Step 1.1.2

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

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

Step 1.1.3

Cancel the common factors.

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

Step 1.2

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

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Step 1.2.1

Reorder $\sin (a)$ and $\csc (a)$ .

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

Step 1.2.2

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

$\frac{1 + \frac{1}{\sin (a)} \sin (a)}{\sec (a)}$

Step 1.2.3

Cancel the common factors.

$\frac{1 + 1}{\sec (a)}$

Step 1.3

Add $1$ and $1$ .

$\frac{2}{\sec (a)}$

Step 2

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

$\frac{2}{\frac{1}{\cos (a)}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$2 \cos (a)$