Final answer:
To simplify the expression tan(theta)sin(2) - 2/(cos(theta)sec(theta)), rewrite sec(theta) as 1/cos(theta). Then, multiply by the reciprocal of cos(theta) to eliminate the fraction. The simplified expression is tan(theta)sin(2) - 2. The cos(theta) in the denominator and the cos(theta) in sec(theta) cancel out, leaving us with simply tan(theta)sin(2) - 2, which is option a).
Step-by-step explanation:
To simplify the expression tan(theta)sin(2) - 2/(cos(theta)sec(theta)), let's rewrite sec(theta) as 1/cos(theta). Now we have tan(theta)sin(2) - 2/(cos(theta)*(1/cos(theta))). Since sec(theta) is the reciprocal of cos(theta), multiplying by sec(theta) is the same as dividing by cos(theta). So the expression becomes tan(theta)sin(2) - 2/(cos(theta)/cos(theta)).
When we divide by a fraction, we can multiply by its reciprocal. So, we can rewrite the expression as tan(theta)sin(2) - 2*(cos(theta)/cos(theta)).
Now, multiplying cos(theta)/cos(theta) will cancel out the denominator, so the expression simplifies to tan(theta)sin(2) - 2.
Simplify the following expression: tan(theta)sin(2) - 2/(cos(theta)sec(-)). To simplify this expression, we start by understanding that sec(-) is the reciprocal of cos(-). Therefore, sec(-) = 1/cos(-). Now, cos(-theta) = cos(theta) because cosine is an even function, which means sec(-theta) = sec(theta).
With that in mind, we substitute sec(theta) back into the original expression to get tan(theta)sin(2) - 2/(cos(theta) * (1/cos(theta))). The cos(theta) in the denominator and the cos(theta) in sec(theta) cancel out, leaving us with simply tan(theta)sin(2) - 2, which is option a).