Questions: Use trigonometric identities to simplify the expression. csc(t) tan(t)

Use trigonometric identities to simplify the expression. csc(t) tan(t)
Transcript text: Use trigonometric identities to simplify the expression. \[ \csc (t) \tan (t) \]
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Solution

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Solution Steps

Step 1: Rewrite the expression using trigonometric identities

Start by expressing \(\csc(t)\) and \(\tan(t)\) in terms of sine and cosine: \[ \csc(t) = \frac{1}{\sin(t)}, \quad \tan(t) = \frac{\sin(t)}{\cos(t)}. \]

Step 2: Substitute the identities into the expression

Substitute the expressions for \(\csc(t)\) and \(\tan(t)\) into the original expression: \[ \csc(t) \tan(t) = \frac{1}{\sin(t)} \cdot \frac{\sin(t)}{\cos(t)}. \]

Step 3: Simplify the expression

Multiply the fractions and simplify: \[ \frac{1}{\sin(t)} \cdot \frac{\sin(t)}{\cos(t)} = \frac{1}{\cos(t)}. \] This simplifies to \(\sec(t)\), since \(\sec(t) = \frac{1}{\cos(t)}\).

Final Answer

\(\boxed{\sec(t)}\)

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