Consider the following two statements. Statement $\boldsymbol{p}$ : The value of $\sin 120^{\circ}$ can be divided by taking $ heta=240^{\circ}$ in the equation $$ 2 \sin \frac{ heta}{2}=\sqrt{1+\sin heta}-\sqrt{1-\sin heta} . $$ Statement $\boldsymbol{q}$ : The angles $A, B, C$ and $D$ of…

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Consider the following two statements. Statement $\boldsymbol{p}$ : The value of $\sin 120^{\circ}$ can be divided by taking $	heta=240^{\circ}$ in the equation $$ 2 \sin \frac{	heta}{2}=\sqrt{1+\sin 	heta}-\sqrt{1-\sin 	heta} . $$ Statement $\boldsymbol{q}$ : The angles $A, B, C$ and $D$ of any quadrilateral $A B C D$ satisfy the equation $$ \cos \left(\frac{1}{2}(A+C)ight)+\cos \left(\frac{1}{2}(B+D)ight)=0 $$ Then the truth values of $p$ and $q$ are respectively.

Aicharya · Accepted Answer

Statement $p$ : $ \sin 120^{\circ}=\cos 30^{\circ}=\frac{\sqrt{3}}{2} \Rightarrow 2 \sin 120^{\circ}=\sqrt{3} $ So, $\sqrt{1+\sin 240^{\circ}}-\sqrt{1-\sin 240^{\circ}}=$ $ \sqrt{\frac{1-\sqrt{3}}{2}}-\sqrt{\frac{1+\sqrt{3}}{2}} 
eq \sqrt{3} $ Statement $\mathbf{q}$ : So, $A+B+C+D=2 \pi$ $ \begin{aligned} &\Rightarrow \frac{A+C}{2}+\frac{B+D}{2}=\pi \ &\Rightarrow \cos \left(\frac{A+C}{2}ight)+\cos \left(\frac{B+D}{2}ight) \ &=\cos \left(\frac{A+C}{2}ight)-\cos \left(\frac{A+C}{2}ight)=0 \end{aligned} $ Therefore, statement $p$ is false and statement $q$ is true

Consider the following two statements. Statement $\boldsymbol{p}$ : The value of $\sin 120^{\circ}$ can be…

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