Write the following in terms of sin(θ) and cos(θ); then simplify if possible.
sec(θ) / tan(θ) =
secØ/tanØ
= (1/cosØ)/(sinØ/cosØ)
= (1/cosØ)(cosØ/sinØ)
= 1/sinØ
To express sec(θ) / tan(θ) in terms of sin(θ) and cos(θ), we need to substitute the definitions of sec(θ) and tan(θ) in terms of sin(θ) and cos(θ):
sec(θ) = 1 / cos(θ)
tan(θ) = sin(θ) / cos(θ)
Replacing these in the expression:
sec(θ) / tan(θ) = (1 / cos(θ)) / (sin(θ) / cos(θ))
Now, multiplying the numerator and denominator by the reciprocal of sin(θ) / cos(θ), which is cos(θ) / sin(θ), we get:
sec(θ) / tan(θ) = (1 / cos(θ)) * (cos(θ) / sin(θ))
Simplifying further:
sec(θ) / tan(θ) = (1 * cos(θ)) / (cos(θ) * sin(θ))
Canceling out the common terms:
sec(θ) / tan(θ) = 1 / sin(θ)
Therefore, sec(θ) / tan(θ) simplifies to 1 / sin(θ), or more commonly written as csc(θ), which stands for cosec(θ).
To write sec(θ) / tan(θ) in terms of sin(θ) and cos(θ), we need to identify the definitions of sec(θ) and tan(θ) in relation to sin(θ) and cos(θ).
1. sec(θ) is the reciprocal of cos(θ), which can be written as 1/cos(θ).
2. tan(θ) is the ratio of sin(θ) to cos(θ), which can be written as sin(θ)/cos(θ).
Therefore, we can rewrite sec(θ) / tan(θ) as:
(1/cos(θ)) / (sin(θ)/cos(θ))
To simplify this expression, we can multiply the numerator by the reciprocal of the denominator. The reciprocal of sin(θ)/cos(θ) is cos(θ)/sin(θ). Thus, we have:
(1/cos(θ)) * (cos(θ)/sin(θ))
The cos(θ) terms cancel out, leaving:
1/sin(θ)
Therefore, sec(θ) / tan(θ) simplifies to 1/sin(θ), or csc(θ).