Add and subtract as indicated. Then simplify your answer if possible. Leave answer in terms of sin(θ) and/or cos(θ). (Remember to enter trigonometric powers such as sin2(x) as (sin(x))2.)
1/sin-sin=
sin is a mathematical operator such as √ , ÷ etc.
by itself it is meaningless, it must have an operand behind it, i.e. sinx or sinØ
(that is like saying: evaluate 1/√ - √ )
1/sinØ - sinØ
= (1 - sin^2 Ø)/(sinØ
= cos^2 Ø/sinØ
which is in terms of sines and cosines as requested
(I would have simplified it to
cotØcosØ )
To simplify the given expression, we need to find a common denominator for the fractions.
The original expression is:
1/sin - sin
Since the first term is a fraction, we can express the second term as a fraction by multiplying it by 1 in the form of sin/sin:
1/sin - sin(1)
Now, we have:
1/sin - sin(1) = 1/sin - sin^2/sin
To add or subtract fractions, we need a common denominator. The common denominator in this case would be sin:
1/sin - sin^2/sin = (1 - sin^2)/sin
Using the identity sin^2(θ) + cos^2(θ) = 1, we can substitute cos^2(θ) for 1 - sin^2(θ):
(1 - sin^2(θ))/sin = cos^2(θ)/sin
Therefore, the simplified expression is cos^2(θ)/sin.
To simplify the expression 1/sin - sin, we'll follow these steps:
Step 1: Find a common denominator for the terms 1/sin and sin.
The common denominator for sin is sin, so we can rewrite 1/sin as sin(θ)/sin(θ).
Step 2: Combine the fractions using the common denominator.
sin(θ)/sin(θ) - sin(θ)
Since we now have the same denominator for both terms, we can subtract the numerators:
(sin(θ) - sin(θ))/sin(θ)
The numerator simplifies to zero:
0/sin(θ)
Step 3: Simplify the expression.
Any expression divided by sin(θ) is equal to zero because sin(θ) cannot be zero. So, the simplified answer is:
0
Therefore, the expression 1/sin - sin simplifies to 0.