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Problem 5 - Entrance Test

If tan A + sec A = 3, then cos A equals

Correct: C

Write tan A + sec A = sin A/cos A + 1/cos A = (1+sin A)/cos A = 3. Let t = tan(A/2). Then (1+2t/(1+t²))/((1-t²)/(1+t²)) = (1+t)²/(1-t²) = (1+t)/(1-t) = 3. Solving 1+t = 3-3t gives t=1/2. Hence cos A = (1-t²)/(1+t²)=3/5.