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 APA
To prove the equation sinθ/(1-cosθ) - 1/(1+cosθ)sinθ = 0, we can start by simplifying the expression step by step:
Step 1: Start with the given expression sinθ/(1-cosθ) - 1/(1+cosθ)sinθ.
Step 2: Simplify the denominator in the first term by multiplying both numerator and denominator by (1+cosθ):
sinθ * (1+cosθ) / [(1-cosθ) * (1+cosθ)] - 1/(1+cosθ)sinθ.
Justification: To eliminate the fraction in the denominator, we multiply the numerator and denominator of the first term by (1+cosθ).
Step 3: Simplify the numerator in the first term:
sinθ + sinθcosθ - 1/[1+cosθ] * sinθ.
Justification: Using the distributive property, we expand the numerator in the first term.
Step 4: Combine like terms in the numerator:
sinθ + sinθcosθ - sinθ/[1+cosθ].
Justification: We have combined the sinθ terms in the numerator.
Step 5: Factor out sinθ in the numerator:
sinθ * [1 + cosθ - 1/(1+cosθ)].
Justification: We factor out sinθ from all the terms in the numerator.
Step 6: Simplify the expression inside the square brackets:
1 + cosθ - 1/(1+cosθ).
Justification: We have combined the terms in the square brackets.
Step 7: Find the common denominator for the third term:
1 + cosθ - (1/cosθ)/(1/cosθ + cosθ/cosθ).
Justification: Using the common denominator rule, we found the common denominator for the third term.
Step 8: Simplify the expression with the common denominator:
1 + cosθ - (1/cosθ)/(1+cosθ).
Justification: We simplified the expression with the common denominator.
Step 9: Combine the fractions in the denominator:
1 + cosθ - 1 / (cosθ + cosθ^2).
Justification: We combined the fractions in the denominator.
Step 10: Simplify the expression with the common denominator:
1 + cosθ - 1 / (1 + cosθ).
Justification: We simplified the expression with the common denominator.
Step 11: Combine the terms in the numerator:
1 + cosθ - 1 = cosθ.
Justification: We combined the terms in the numerator.
Step 12: Now the equation becomes:
sinθ * cosθ = 0.
Justification: We substitute the simplified numerator into the equation.
Step 13: Multiply both sides of the equation by 1/sinθ:
cosθ = 0.
Justification: We multiplied both sides of the equation by 1/sinθ to eliminate sinθ from the equation.
Step 14: The equation is satisfied when cosθ equals zero.
Justification: From Step 13, we found that cosθ = 0 is the solution to the equation.
Therefore, we have proven that sinθ/(1-cosθ) - 1/(1+cosθ)sinθ = 0 when cosθ = 0.
To prove the given equation:
sinθ / (1 - cosθ) - 1 / (1 + cosθ) sinθ = 0
Step 1: Combine the fractions
[(sinθ * (1 + cosθ)) - (1 - cosθ)] / [(1 - cosθ) * (1 + cosθ)] sinθ = 0
Step 2: Simplify the numerator
[(sinθ + sinθcosθ) - 1 + cosθ] / [(1 - cosθ) * (1 + cosθ)] sinθ = 0
Step 3: Combine like terms in the numerator
[sinθ + sinθcosθ - 1 + cosθ] / [(1 - cosθ) * (1 + cosθ)] sinθ = 0
Step 4: Rearrange the terms in the numerator
[(sinθ - 1) + sinθcosθ + cosθ] / [(1 - cosθ) * (1 + cosθ)] sinθ = 0
Step 5: Combine like terms in the numerator
[(2sinθ + sinθcosθ + cosθ - 1)] / [(1 - cosθ) * (1 + cosθ)] sinθ = 0
Step 6: Factor out sinθ from the numerator
[2sinθ + sinθ(cosθ + 1) + (cosθ - 1)] / [(1 - cosθ) * (1 + cosθ)] sinθ = 0
Step 7: Combine like terms in the numerator
[2sinθ + sinθ(cosθ + 1) + (cosθ - 1)] / [(1 - cosθ) * (1 + cosθ)] sinθ = 0
Step 8: Simplify the denominator
[(1 - cosθ)(1 + cosθ)] = sinθ^2
[(1 - cosθ^2)] = sinθ^2
1 - cos^2θ = sin^2θ
Step 9: Substitute the simplification of the denominator back into the equation
[2sinθ + sinθ(cosθ + 1) + (cosθ - 1)] / (1 - cos^2θ) sinθ = 0
Step 10: Simplify the equation
[2sinθ + sinθ(cosθ + 1) + (cosθ - 1)] / sinθ = 0
Step 11: Cancel out sinθ in the numerator and denominator
2 + cosθ + 1 + cosθ - 1 = 0
2cosθ + cosθ = 0
3cosθ = 0
Step 12: Divide both sides by 3
cosθ = 0
Step 13: We know that sinθ/(1 - cosθ) - 1/(1 + cosθ) sinθ = 0 is true if cosθ = 0.
Justification for each step:
1. Combining the fractions: We combine the fractions to obtain a single fraction.
2. Simplifying the numerator: We simplify the expression in the numerator by distributing sinθ.
3. Combining like terms in the numerator: We combine the terms that have sinθ.
4. Rearranging the terms in the numerator: We rearrange the terms to group sinθ together.
5. Combining like terms in the numerator: We combine the terms that don't have sinθ.
6. Factoring out sinθ from the numerator: We factor out sinθ to simplify the expression.
7. Combining like terms in the numerator: We combine the terms that have sinθ.
8. Simplifying the denominator: We simplify the denominator using the trigonometric identity sin^2θ = 1 - cos^2θ.
9. Substituting the simplification of the denominator back into the equation: We substitute the simplified denominator back into the equation.
10. Simplifying the equation: We simplify the equation with the substituted denominator.
11. Canceling out sinθ in the numerator and denominator: We cancel out sinθ to simplify the equation.
12. Dividing both sides by 3: We divide both sides by 3 to solve for cosθ.
13. Identifying the condition for the equation to be true: We find that the equation is true if cosθ = 0.
References:
None used.