Prove that : sinθ+cosθ−1sinθ−cosθ+1=secθ−tanθ1.
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Step-by-Step Solution
Step 1: Start with the Left Hand Side (LHS)
We begin by considering the Left Hand Side (LHS) of the given identity. Our goal is to manipulate this expression algebraically until it matches the Right Hand Side (RHS).
Step 2: Divide numerator and denominator by cosθ
To introduce tanθ and secθ into the expression, we divide every term in both the numerator and the denominator by cosθ. This is a common technique when dealing with trigonometric identities involving sinθ and cosθ to transform them into tanθ and secθ.
Step 3: Rearrange terms and use identity
We rearrange the terms in the numerator to group tanθ+secθ. Then, we replace the constant '1' in the numerator with the identity sec2θ−tan2θ=1. This step is crucial for factoring later.
Step 4: Factor the numerator
We factor the term (sec2θ−tan2θ) using the difference of squares formula, a2−b2=(a−b)(a+b). Then, we factor out the common term (tanθ+secθ) from the numerator.
Step 5: Simplify the expression
Notice that the term (1−secθ+tanθ) in the numerator is identical to the denominator (tanθ−secθ+1). We can cancel these terms, simplifying the expression to tanθ+secθ.
Step 6: Convert to the form of RHS
Finally, we multiply the numerator and denominator by (secθ−tanθ). Using the identity sec2θ−tan2θ=1, the numerator simplifies to 1, which matches the Right Hand Side (RHS) of the original identity.