Write the following in terms of sin(θ) and cos(θ); then simplify if possible.
csc(θ) - cot(θ)cos(θ) =
csc(θ) - cot(θ)cos(θ)
= 1/sinØ - (cosØ/sinØ)cosØ
= (1- cos^2 Ø)/sinØ
= sin^2 Ø/sinØ
= sinØ
To rewrite csc(θ) - cot(θ)cos(θ) in terms of sin(θ) and cos(θ), we need to know the definitions of the trigonometric functions csc and cot:
csc(θ) = 1/sin(θ)
cot(θ) = cos(θ)/sin(θ)
Now we can substitute these definitions into the expression:
csc(θ) - cot(θ)cos(θ)
= 1/sin(θ) - (cos(θ)/sin(θ)) * cos(θ)
= 1/sin(θ) - (cos(θ) * cos(θ))/sin(θ)
= 1/sin(θ) - (cos²(θ))/sin(θ)
To combine these fractions into a single fraction, we need a common denominator. The common denominator is sin(θ). So, we have:
= (1 - cos²(θ))/sin(θ)
Using the identity sin²(θ) + cos²(θ) = 1, we can rewrite 1 - cos²(θ) as sin²(θ). Thus, the expression simplifies to:
= sin²(θ)/sin(θ)
= sin(θ)
Therefore, csc(θ) - cot(θ)cos(θ) simplifies to sin(θ).