Question Number 207128 by hardmath last updated on 07/May/24
$$\mathrm{log}_{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\mathrm{x}}} \:\:\mathrm{sinx}\:\:−\:\:\frac{\mathrm{1}}{\mathrm{log}_{\boldsymbol{\mathrm{cos}}\:\boldsymbol{\mathrm{x}}} \:\:\mathrm{tanx}}\:\:=\:\:? \\ $$
Answered by Frix last updated on 07/May/24
$$=\frac{\mathrm{ln}\:{s}}{\mathrm{ln}\:{t}}−\frac{\mathrm{ln}\:{c}}{\mathrm{ln}\:{t}}=\frac{\mathrm{ln}\:{s}\:−\mathrm{ln}\:{c}}{\mathrm{ln}\:\frac{{s}}{{c}}} \\ $$$$\mathrm{If}\:{c}>\mathrm{0}\wedge{s}>\mathrm{0}\:\mathrm{this}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{1},\:\mathrm{else}\:\mathrm{it}'\mathrm{s}\:\mathrm{getting} \\ $$$$\mathrm{complicated}. \\ $$
Commented by hardmath last updated on 07/May/24
$$\mathrm{yes}\:\mathrm{dear}\:\mathrm{professor}\:>\mathrm{0} \\ $$
Commented by Frix last updated on 07/May/24
$${c}>\mathrm{0}\wedge{s}>\mathrm{0}\:\Rightarrow\:\mathrm{ln}\:\frac{{s}}{{c}}\:=\mathrm{ln}\:{s}\:−\mathrm{ln}\:{c} \\ $$
Commented by hardmath last updated on 07/May/24
$$\mathrm{Answer}\:=\:\mathrm{1}\:.? \\ $$
Commented by Frix last updated on 07/May/24
$$\mathrm{Yes}\:\mathrm{for}\:\mathrm{cos}\:{x}\:>\mathrm{0}\:\wedge\mathrm{sin}\:{x}\:>\mathrm{0}\:\wedge\mathrm{cos}\:{x}\:\neq\mathrm{sin}\:{x} \\ $$