Questions: Simplify: 10 sin (-t) csc (-t)+6 cos (-t) sec (-t) A) 16 B) 2 C) 4 D) -16 E) -4 F) None of these

Solution Steps

To simplify the given expression, we need to use the properties of trigonometric functions. Specifically, we will use the fact that \(\sin(-t) = -\sin(t)\) and \(\cos(-t) = \cos(t)\). Additionally, we will use the identities \(\csc(t) = \frac{1}{\sin(t)}\) and \(\sec(t) = \frac{1}{\cos(t)}\).

1. Substitute \(\sin(-t)\) and \(\cos(-t)\) with their equivalent expressions.
2. Simplify the trigonometric functions using their reciprocal identities.
3. Combine and simplify the resulting expression.

We start by substituting \(\sin(-t)\) and \(\cos(-t)\) with their equivalent expressions: \[ \sin(-t) = -\sin(t) \quad \text{and} \quad \cos(-t) = \cos(t) \]

Next, we use the reciprocal identities: \[ \csc(t) = \frac{1}{\sin(t)} \quad \text{and} \quad \sec(t) = \frac{1}{\cos(t)} \]

Substituting these into the original expression: \[ 10 \sin(-t) \csc(-t) + 6 \cos(-t) \sec(-t) \] becomes: \[ 10 (-\sin(t)) \csc(t) + 6 \cos(t) \sec(t) \] which simplifies to: \[ 10 (-\sin(t)) \left(\frac{1}{\sin(t)}\right) + 6 \cos(t) \left(\frac{1}{\cos(t)}\right) \] This further simplifies to: \[ 10 (-1) + 6 (1) = -10 + 6 = -4 \]

Final Answer