Problem 30
Question
Verify each identity. $$1-\frac{\cos ^{2} x}{1+\sin x}=\sin x$$
Step-by-Step Solution
Verified Answer
Yes, the original equation \(1-\frac{\cos ^{2} x}{1+\sin x} = \sin x\) is a valid identity as it simplifies to \(\sin x = \sin x\).
1Step 1: Replace with Identity.
Replace \(\cos^{2}x\) with the Pythagorean Identity \(\cos^{2}x = 1 - \sin^{2}x\). This results in the equation: \(1-\frac{1-\sin^{2} x}{1+\sin x}\)
2Step 2: Simplify the Complex Fraction.
Multiply the denominator by \((1-\sin x)\) to clear the complex fraction. Now our equation: \(1-(1-\sin x) = \sin x\)
3Step 3: Simplify the Expressions.
Simplify both sides of the equation. After removing the brackets and cancelling, we are left with: \(\sin x = \sin x\)
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