Question

If $b>1,\sin t>0,\cos t>0$ and ${\log }_b\left(\sin t\right)=x$ then ${\log }_b\left(\cos t\right)$ is equal to

• A. $b^{x^2}$
• B. $2{\log }_b(1-b^{x/2})$
• C. ${\log }_b\left(\sqrt{1-b^{2x}}\right)$
• D. $\sqrt{1-x^2}$

Answer 1

The correct option is C ${\log }_b\left(\sqrt{1-b^{2x}}\right)$

${\log }_b\left(\sin t\right)=x$

If ${\log }_{a}b=\gamma \Rightarrow b=a\gamma$

Similarly, $\sin t=b^x$

${\sin }^{2}t+{\cos }^{2}t=1$

or, ${\cos }^{2}t=1-\sin t=1-(b^x{)}^2$

$\cos t=\sqrt{1-b^{2}x}$

${\log }_b\left(\cos t\right)={\log }_b\sqrt{1-b^{2x}}$