Prove
sin
1 cos
1 cos
sin
0
1. Show each step of your proof.
2. Provide written justification for each step of your proof.
C. If you use sources, include all in-text citations and references in
To prove the given expression, we need to transform it step by step until we obtain a valid equation or identity.
1. Start with the expression:
sin(θ) / (1 - cos(θ)) - 1 / (1 + cos(θ)) * sin(θ) = 0
2. Now, let's simplify the expression. To do that, we need to find a common denominator for the two fractions:
((sin(θ) * (1 + cos(θ))) - (1 - cos(θ)) * sin(θ)) / ((1 - cos(θ)) * (1 + cos(θ))) = 0
3. Simplify the numerator by expanding both terms:
(sin(θ) + sin(θ)*cos(θ) - sin(θ) + sin(θ)*cos(θ)) / ((1 - cos(θ)) * (1 + cos(θ))) = 0
4. Combine similar terms in the numerator:
(2sin(θ)*cos(θ)) / ((1 - cos(θ)) * (1 + cos(θ))) = 0
5. Now, let's analyze the expression. For this equation to be true, either the numerator must be zero, or the denominator must be zero.
a) Numerator:
2sin(θ)*cos(θ) = 0
This equation is satisfied when either sin(θ) = 0 or cos(θ) = 0.
b) Denominator:
(1 - cos(θ)) * (1 + cos(θ)) = 0
This equation is satisfied when either (1 - cos(θ)) = 0 or (1 + cos(θ)) = 0.
6. Analyzing the possibilities:
a) sin(θ) = 0
If sin(θ) = 0, then the whole expression becomes 0/((1 - cos(θ)) * (1 + cos(θ))), which equals zero. Therefore, this case satisfies the equation.
b) cos(θ) = 0
If cos(θ) = 0, then the whole expression becomes (2sin(θ)*0) / ((1 - 0) * (1 + 0)), which is also zero. Hence, this case satisfies the equation.
c) (1 - cos(θ)) = 0
If (1 - cos(θ)) = 0, then cos(θ) = 1. Plugging this value into the original expression, we get sin(θ) / (1 - 1) - 1 / (1 + 1) * sin(θ). This simplifies to sin(θ) / 0 - 1 / 2 * sin(θ), which is not defined. Therefore, this case does not satisfy the equation.
d) (1 + cos(θ)) = 0
If (1 + cos(θ)) = 0, then cos(θ) = -1. Plugging this value into the original expression, we get sin(θ) / (1 + 1) - 1 / (1 - 1) * sin(θ). This simplifies to sin(θ) / 2 - 1 / 0 * sin(θ), which is also not defined. Hence, this case does not satisfy the equation.
7. The only cases that satisfy the equation are when sin(θ) = 0 or cos(θ) = 0.
Therefore, the given expression sin(θ) / (1 - cos(θ)) - 1 / (1 + cos(θ)) * sin(θ) = 0 is true when sin(θ) = 0 or cos(θ) = 0.
8. Justification for each step:
- Step 1: We start with the given expression.
- Steps 2-4: We simplify and combine terms using common denominator techniques and basic algebraic manipulation of fractions.
- Step 5: We analyze and identify the possible cases that could satisfy the equation.
- Steps 6-7: We evaluate each case by substituting the values back into the original expression and determining if it is valid or not.
- Step 8: We provide a brief explanation of each step's purpose in the solution process.
Remember that it's always important to check for potential restrictions in the denominator to ensure that the expression is well-defined.