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Question Number 207128 by hardmath last updated on 07/May/24

$${\log }_{\mathbf{tan}\mathbf{x}}\mathrm{sinx}-\frac{1}{{\log }_{\mathbf{cos}\mathbf{x}}\mathrm{tanx}}=?$$

Answered by Frix last updated on 07/May/24

$$=\frac{\ln s}{\ln t}-\frac{\ln c}{\ln t}=\frac{\ln s-\ln c}{\ln \frac{s}{c}}$$$$\mathrm{If}c>0\wedge s>0\mathrm{this}\mathrm{is}\mathrm{equal}\mathrm{to}1,\mathrm{else}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{getting}$$$$\mathrm{complicated}.$$

Commented by hardmath last updated on 07/May/24

$$\mathrm{yes}\mathrm{dear}\mathrm{professor}>0$$

Commented by Frix last updated on 07/May/24

$$c>0\wedge s>0\Rightarrow \ln \frac{s}{c}=\ln s-\ln c$$

Commented by hardmath last updated on 07/May/24

$$\mathrm{Answer}=1.?$$

Commented by Frix last updated on 07/May/24

$$\mathrm{Yes}\mathrm{for}\cos x>0\wedge \sin x>0\wedge \cos x\ne \sin x$$

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