Question

$\tan A=a\tan B,\sin A=b\sin B\Rightarrow \frac{b^2-1}{a^2-1}$ is equal to

• A. ${\sin }^{2}A$
• B. ${\sin }^{3}A$
• C. ${\cos }^{2}A$
• D. ${\cos }^{3}A$

Answer 1

Answer: (C)

${\cos }^{2}A$

$\tan A=a\tan B$ and $\sin A=b\sin B$

$\frac{\sin A}{\cos A}=a\frac{\sin B}{\cos B}$

$\frac{b\sin B}{\cos A}=\frac{a\sin B}{\cos B}$...... Given, $b\sin B=\sin A$

$\frac{b}{\cos A}=\frac{a}{\cos B}\Rightarrow \cos B=\frac{a}{b}\cos A$

${\sin }^{2}A=b^2{\sin }^{2}B\Rightarrow 1-{\cos }^{2}A=b^2(1-{\cos }^{2}B)$

$\Rightarrow 1-{\cos }^{2}A=b^2(1-\frac{a^2}{b^2}{\cos }^{2}A)$

$\Rightarrow 1-{\cos }^{2}A=b^2-a^2{\cos }^{2}A$

$\Rightarrow {\cos }^{2}A(a^2-1)=b^2-1$

$\Rightarrow \frac{(b^2-1)}{(a^2-1)}={\cos }^{2}A$