| Direction: For the following questions, choose the correct answer from the codes [a], [b], [c] and [d] defined as follows. |
| Consider the identify \[\frac{\sin \,\frac{\theta }{2}-\sin \,\frac{\phi }{2}}{\cos \,\frac{\theta }{2}+\cos \,\frac{\theta }{2}}=\tan \,\frac{\theta -\phi }{4}\]. |
| Statement I \[{{\left( \frac{\cos A+\cos B}{\sin A-\sin B} \right)}^{n}}+{{\left( \frac{sinA+sinB}{\cos A-\cos B} \right)}^{n}}\] \[=\left\{ \begin{matrix} 2{{\cot }^{n}}\frac{A-B}{2}, & \text{if}\,\text{n}\,\text{is}\,\text{odd} \\ 0, & \text{if}\,\text{n}\,\text{is}\,\text{seven} \\ \end{matrix} \right.\] |
| Statement II \[\frac{\cos \,A+\cos \,B}{\sin \,A-\sin \,B}=\cot \,\frac{A-B}{2}\]. |
A) Statement I is true. Statement II is also true and Statement II is the correct explanation of Statement I.
B) Statement I is true. Statement II is also true and Statement II is not the correct explanation of Statement I.
C) Statement I is true. Statement II is false.
D) Statement I is false. Statement II is true.
Correct Answer: D
Solution :
\[\because \] \[\left( \frac{\cos \,A+\cos \,B}{\sin \,A-\sin \,B} \right)+\,{{\left( \frac{\sin \,A+\sin \,B}{\cos \,A-\cos \,B} \right)}^{n}}\] \[=\left( \frac{2\,\cos \,\frac{A+B}{2}\cdot \,\cos \,\frac{A-B}{2}}{2\,\cos \,\frac{A+B}{2}\cdot \,\sin \,\frac{A-B}{2}} \right)\] \[={{\cot }^{n}}\frac{A-B}{2}+{{\cot }^{n}}\frac{B-A}{2}\] Now, \[{{\cot }^{n}}\,\frac{A-B}{2}+{{\cot }^{n}}\frac{B-A}{2}\]\[\left\{ \begin{matrix} 0, & if\,n\,is\,odd \\ 2\,{{\cot }^{n}}\frac{A-B}{2}, & if\,n\,is\,even \\ \end{matrix} \right.\]
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