If $\log_{(cos x)}(sin x) + \log_{(sin x)}(cos x) = 2,$ then the value of $x$ is

Question and options

\log_{(cos x)}(sin x) + \log_{(sin x)}(cos x) = 2,

then the value of

x

$n\pi + \dfrac{\pi}{4}, n$ is an integer
$2n\pi + \dfrac{\pi}{4}, n$ is an integer
$\dfrac{n\pi}{4}, n$ is an integer
$\dfrac{n\pi}{4} + \dfrac{\pi}{4}, n$ is an integer

Answer and explanation

Correct answer: B - $2n\pi + \dfrac{\pi}{4}, n$ is an integer

With t = log_cos(sin x), t+1/t=2 gives t=1, so sin x = cos x, with both positive and not 1 (valid log bases/arguments). So x is in the first quadrant: x = 2n pi + pi/4 (n pi + pi/4 with odd n gives negative sin and cos).

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