Question : If $\sqrt{1+\frac{\sqrt{3}}{2}}-\sqrt{1-\frac{\sqrt{3}}{2}}=c$, then the value of $\mathrm{c}$ is:
Option 1: 1
Option 2: 4
Option 3: 3
Option 4: 2
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Correct Answer: 1
Solution :
Given: $\sqrt{1+\frac{\sqrt{3}}{2}}-\sqrt{1-\frac{\sqrt{3}}{2}}=c$
Squaring both sides, we get
⇒ $\left (\sqrt{1+\frac{\sqrt{3}}{2}}-\sqrt{1-\frac{\sqrt{3}}{2}}\right ) ^2=c^2$
⇒ $1+\frac{\sqrt{3}}{2}+1-\frac{\sqrt{3}}{2}-2\times\sqrt{1-\frac{3}{4}}=c^2$
⇒ $2-2\times\sqrt{\frac{1}{4}}=c^2$
⇒ $2-2\times\frac{1}{2}=c^2$
⇒ $2-1=c^2$
⇒ $c=1$
Hence, the correct answer is 1.
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