<p>Let <img alt="$A$" height="14" src="https://latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png" width="15" /> be an acute angle such that <img alt="$\tan A = 2 \cos A.$" height="14" src="https://latex.artofproblemsolving.com/7/7/4/77450b83e9f88b46bf09a7385fc47b2e0934aa06.png" width="127" /> Find the number of positive integers <img alt="$n$" height="10" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="13" /> less than or equal to <img alt="$1000$" height="14" src="https://latex.artofproblemsolving.com/6/e/e/6ee927e1332358c96c62c277441c907c4f51057f.png" width="38" /> such that <img alt="$\sec^n A + \tan^n A$" height="16" src="https://latex.artofproblemsolving.com/5/4/c/54ccd6ec1d374e004281fa707a6f581c077caa10.png" width="125" /> is a positive integer whose units digit is <img alt="$9.$" height="14" src="https://latex.artofproblemsolving.com/f/6/a/f6a1b9d5cec741422bfb37f91db47fd7514f5377.png" width="16" /></p>

Question No: 447

Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$

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Tags: Trigonometry

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