Use trigonometric identities to simplify the expression.csc(−α)tan(α)

Sure, let's simplify the expression csc(−α)tan(α) using trigonometric identities.

Step 1: Write down the given expression. csc(−α)tan(α)

Step 2: Replace csc(−α) and tan(α) with their equivalent expressions in terms of sine and cosine. Remember that csc(α) = 1/sin(α) and tan(α) = sin(α)/cos(α). Also, note that csc(−α) = -csc(α) because cosecant is an odd function.

So, csc(−α)tan(α) = -1/sin(α) * sin(α)/cos(α)

Step 3: Simplify the expression. The sin(α) in the numerator and denominator will cancel out.

So, -1/sin(α) * sin(α)/cos(α) = -1/cos(α)

Step 4: Replace -1/cos(α) with its equivalent trigonometric function. Remember that sec(α) = 1/cos(α).

So, -1/cos(α) = -sec(α)

Therefore, csc(−α)tan(α) simplifies to -sec(α).