Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (t)}$ .

$\sin (t) \sin (t) + \frac{\cos (t)}{\sec (t)}$

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Add $1$ and $1$ .

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

Step 1.4

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

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

Step 1.5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (t)}$ .

$\sin^{2} (t) + \cos (t) \cos (t)$

Step 1.6

Simplify.

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.6.3

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

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

Step 1.6.4

Add $1$ and $1$ .

$\sin^{2} (t) + \cos^{2} (t)$

Step 2

Apply pythagorean identity.

$1$