If b>1, sin x>0, cos x>0 and log_bsin x=a, then log_bcos x equals

1980 AMC 12 Problem 18

Attempt Problem 18 of the 1980 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1980 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

18.

b\gt1,

\sin x\gt0,

\cos x\gt0

and

\log_b\sin x=a,

then

\log_b\cos x

equals

2\log_b(1-b^{\frac{a}{2}})

\sqrt{1-a^2}

b^{a^2}

\frac12\log_b(1-b^{2a})

none of these

Convert

\log_b\sin x=a

into an exponential equation

Use

\cos x=\sqrt{1-\sin^2x}

and the positivity of

\cos x

We have

\sin x=b^a,

\cos x=\sqrt{1-b^{2a}}.

Therefore

\log_b\cos x =\frac12\log_b(1-b^{2a}).

Therefore, the correct answer is D.