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QJEEAdv03Maf351: JEE Advanced 2003 Mathematics - Class 11 JEEAdv03Maf351 Solution: If is the area of sided regular polygon inscribed in a circle…

If In{I_n} is the area of nn sided regular polygon inscribed in a circle of unit radius and On{O_n} be the area of the polygon circumscribing the given circle, prove that </mo></mrow><mrowdata−mjx−texclass="ORD"><msub><mi>I</mi><mi>n</mi></msub></mrow><mo>=</mo><mrowdata−mjx−texclass="ORD"><mfrac><mrowdata−mjx−texclass="ORD"><mrowdata−mjx−texclass="ORD"><msub><mi>O</mi><mi>n</mi></msub></mrow></mrow><mn>2</mn></mfrac></mrow><mrowdata−mjx−texclass="INNER"><modata−mjx−texclass="OPEN">(</mo><mrowdata−mjx−texclass="ORD"><mn>1</mn><mo>+</mo><msqrt><mn>1</mn><mo>−</mo><mrowdata−mjx−texclass="ORD"><msup><mrowdata−mjx−texclass="ORD"><mrowdata−mjx−texclass="INNER"><modata−mjx−texclass="OPEN">(</mo><mrowdata−mjx−texclass="ORD"><mrowdata−mjx−texclass="ORD"><mfrac><mrowdata−mjx−texclass="ORD"><mn>2</mn><mrowdata−mjx−texclass="ORD"><msub><mi>I</mi><mi>n</mi></msub></mrow></mrow><mi>n</mi></mfrac></mrow></mrow><modata−mjx−texclass="CLOSE">)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow><modata−mjx−texclass="CLOSE">)</mo></mrow></math></mjx−assistive−mml></mjx−container></mjx−assistive−mml></mjx−container></mjx−assistive−mml></mjx−container></mo></mrow><mrow data-mjx-texclass="ORD"><msub><mi>I</mi><mi>n</mi></msub></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msub><mi>O</mi><mi>n</mi></msub></mrow></mrow><mn>2</mn></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>+</mo><msqrt><mn>1</mn><mo>−</mo><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>2</mn><mrow data-mjx-texclass="ORD"><msub><mi>I</mi><mi>n</mi></msub></mrow></mrow><mi>n</mi></mfrac></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container>

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