Copper is extracted from sulphide ore using the method

Let a1 = 1, a2, a3, a4, ........ be consecutive natural numbers. Then&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>tan</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⁡</mo><mfenced separators="|"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn>1</mn></msub><msub><mi mathvariant="normal">a</mi><mn>2</mn></msub></mrow></mfrac></mfenced><mo>+</mo><msup><mi>tan</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⁡</mo><mfenced separators="|"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn>2</mn></msub><msub><mi mathvariant="normal">a</mi><mn>3</mn></msub></mrow></mfrac></mfenced><mo>+</mo><mo>…</mo><mo>.</mo><mo>.</mo><mo>+</mo><msup><mi>tan</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⁡</mo><mfenced separators="|"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn>2021</mn></msub><msub><mi mathvariant="normal">a</mi><mn>2022</mn></msub></mrow></mfrac></mfenced></math>&nbsp;is equal to

If&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>x</mi><mn>3</mn></msub><mo>,</mo><msub><mi>x</mi><mn>4</mn></msub></math>&nbsp; are the roots of the equation&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><msup><mi>x</mi><mn>3</mn></msup><mi>sin</mi><mn>2</mn><mi>β</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mi>cos</mi><mn>2</mn><mi>β</mi><mo>−</mo><mi>x</mi><mi>cos</mi><mi>β</mi><mo>−</mo><mi>sin</mi><mi>β</mi><mo>=</mo></math>,&nbsp; then the value of&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover><msup><mi>tan</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>x</mi><mi>i</mi></msub></math>&nbsp;is

<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∑</mo><mrow><mi mathvariant="normal">r</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">n</mi></msubsup><mo> </mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>&nbsp;</mo><mfenced separators="|"><mfrac><mrow><msqrt><mi mathvariant="normal">r</mi></msqrt><mo>-</mo><msqrt><mi mathvariant="normal">r</mi><mo>-</mo><mn>1</mn></msqrt></mrow><msqrt><mi mathvariant="normal">r</mi><mo>(</mo><mi mathvariant="normal">r</mi><mo>+</mo><mn>1</mn><mo>)</mo></msqrt></mfrac></mfenced></math>&nbsp;is equal to

Copper is extracted from sulphide ore using the method

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