Math Problem Statement
prove (sintheta/(1+costheta))+(1+costheta)/(sintheta)=2csctheta
Solution
We are tasked with proving the identity:
Step 1: Combine the terms on the left-hand side (LHS)
We start by simplifying the left-hand side (LHS) into a single fraction. To do this, we need a common denominator for the two fractions:
The common denominator will be:
So, we rewrite each term with this common denominator:
Now combine the two terms:
Step 2: Simplify the numerator
Expand the square in the numerator:
So the numerator becomes:
Now use the Pythagorean identity :
Thus, the LHS is now:
Step 3: Factor the numerator
Factor out the common factor of 2:
Now, cancel from the numerator and denominator:
Step 4: Final simplification
We know that , so:
Thus, we have:
This completes the proof.
Would you like to go over any specific details, or do you have further questions?
Related Questions:
- Can you prove the identity ?
- How do trigonometric identities simplify expressions involving and ?
- What is the proof for ?
- How can we use the Pythagorean identity to simplify more complicated trigonometric expressions?
- What is the method for deriving sum and difference formulas for trigonometric functions?
Tip:
Remember that the Pythagorean identity, , is very useful when simplifying trigonometric expressions or proving identities.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Trigonometric Identities
Simplification of Expressions
Formulas
sin^2(θ) + cos^2(θ) = 1
csc(θ) = 1/sin(θ)
Theorems
Pythagorean Identity
Suitable Grade Level
Grades 10-12
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