Prove : (sinA-cosA+1)/(sinA+cosA-1) = (1+sinA)/cosA
multiply top and bottom of the left side by
(sinA + cosA + 1) and expand carefully
remember sin^2A + cos^2A = 1
and in the bottom replace the
-cos^2A with (1-sin^2A)
To prove the given equation:
(sinA - cosA + 1) / (sinA + cosA - 1) = (1 + sinA) / cosA
We need to manipulate the left side of the equation to match the right side.
Step 1: Simplify both the numerator and denominator.
Numerator:
sinA - cosA + 1
Denominator:
sinA + cosA - 1
Step 2: Multiply both the numerator and denominator by the conjugate of the denominator, which is (sinA + cosA + 1).
Numerator:
(sinA - cosA + 1) * (sinA + cosA + 1) = sin^2A + cos^2A + 1
Denominator:
(sinA + cosA - 1) * (sinA + cosA + 1) = sin^2A + cos^2A
Step 3: Simplify the numerator and denominator.
Numerator:
sin^2A + cos^2A + 1 = 1 + (sin^2A + cos^2A) = 1 + 1 = 2
Denominator:
sin^2A + cos^2A
Step 4: Rewrite the equation.
2 / (sin^2A + cos^2A) = (1 + sinA) / cosA
Step 5: Recall the identity sin^2A + cos^2A = 1.
2 / 1 = (1 + sinA) / cosA
Step 6: Simplify further.
2 = (1 + sinA) / cosA
Therefore, the given equation is proven.