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Which trigonometric identity is used directly to prove that $\sec A(1-\sin A)(\sec A+\tan A)=1$?
A$\sin^2 A+\cos^2 A=1$
B$1+\cot^2 A=\sec^2 A$
C$\tan^2 A+\cot^2 A=1$
D$\sec^2 A+\tan^2 A=1$
Answer & Solution
Correct answer: A. $\sin^2 A+\cos^2 A=1$
After writing $\sec A=\frac{1}{\cos A}$ and $\tan A=\frac{\sin A}{\cos A}$, the expression becomes $\frac{(1-\sin A)(1+\sin A)}{\cos^2 A}=\frac{1-\sin^2 A}{\cos^2 A}$. Then $1-\sin^2 A=\cos^2 A$ follows from $\sin^2 A+\cos^2 A=1$, so the value is 1.
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