Prove the identity: (sin θ)/(1 + cos θ) + (1 + cos θ)/(sin θ) = 2 cosec θ.
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Step-by-Step Solution
Step 1: Combine fractions on the Left Hand Side
To add the two fractions on the Left Hand Side (LHS), we find a common denominator, which is the product of the individual denominators: sinθ(1+cosθ). We then cross-multiply the numerators and add them.
Step 2: Expand the numerator
Next, we expand the term (1+cosθ)2 in the numerator using the algebraic identity (a+b)2=a2+2ab+b2. Here, a=1 and b=cosθ.
Step 3: Apply Pythagorean Identity
We know the fundamental trigonometric identity sin2θ+cos2θ=1. We substitute this into the numerator, which simplifies the expression.
Step 4: Factor and simplify
We factor out 2 from the numerator. Since (1+cosθ) appears in both the numerator and the denominator, we can cancel this common term, simplifying the fraction.
Step 5: Express in terms of cosecant
Finally, we use the reciprocal identity cosecθ=sinθ1 to express the simplified LHS in terms of cosecant, which matches the Right Hand Side (RHS) of the identity.