Question

If a sin 45°= b cosec 30°, then the value of \(\frac {a^4}{b^4}\) is

• A. 1
• B. 2⁸
• C. 2³
• D. 2⁶

Answer 1

The Correct Option is D

We are given that $a \sin 45^\circ = b \csc 30^\circ$. 
We know that $\sin 45^\circ = \frac{1}{\sqrt{2}}$ and $\csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{1/2} = 2$. 
Substituting these values into the equation, we get $$ a \cdot \frac{1}{\sqrt{2}} = b \cdot 2 $$ $$ \frac{a}{\sqrt{2}} = 2b $$ $$ a = 2b \sqrt{2} $$ $$ \frac{a}{b} = 2 \sqrt{2} $$ Now we want to find $\frac{a^4}{b^4}$. We have $$ \frac{a^4}{b^4} = \left(\frac{a}{b}\right)^4 = (2 \sqrt{2})^4 = (2^1 \cdot 2^{1/2})^4 = (2^{3/2})^4 = 2^{3/2 \cdot 4} = 2^6 = 64 $$ Therefore, $\frac{a^4}{b^4} = 64$.