First slide

Trigonometric Identities

Question

. It b > 1, sin t > 0, cost > 0 and $\log_{b} (\sin t) = x$ then $\log_{b} (\cos t)$ is equal to

Moderate

A

$\frac{1}{2} \log_{b} (1 - b^{2 X})$
B

$2 \log (1 - b^{x / 2})$
C

$\log_{b} \sqrt{1 - b^{2 x}}$
D

$\sqrt{1 - x^{2}}$

Solution

$\begin{array}{l} \log_{b} \sin t = x or \sin t = b^{x} \\ or \log_{b} (\cos t) = y, then b^{y} = \cos t \\ or b^{2 y} = \cos^{2} t = 1 - \sin^{2} t = 1 - b^{2 x} \\ or 2 y = \log_{b} (1 - b^{2 x}) \\ or y = \frac{1}{2} \log_{b} (1 - b^{2 x}) = \log_{b} \sqrt{1 - b^{2 x}} \end{array}$

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