<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi mathvariant="normal">x</mi><mo>)</mo><msup><mi>cosec</mi><mn>2</mn></msup><mo>⁡</mo><mi>xdx</mi></math>&nbsp;is equal to

If&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi mathvariant="normal">p</mi><mo>→</mo></mover></math>and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi mathvariant="normal">q</mi><mo>→</mo></mover></math>are two diagonals of &nbsp;a quadrilateral such that&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mrow><mover><mi mathvariant="normal">p</mi><mo>→</mo></mover><mo>-</mo><mover><mi mathvariant="normal">q</mi><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mover><mi mathvariant="normal">p</mi><mo>→</mo></mover><mo>.</mo><mover><mi mathvariant="normal">q</mi><mo>→</mo></mover></math>,&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mover><mi mathvariant="normal">p</mi><mo>→</mo></mover></mfenced><mo>=</mo><mn>1</mn><mo>,</mo><mfenced close="|" open="|"><mover><mi mathvariant="normal">q</mi><mo>→</mo></mover></mfenced><mo>=</mo><msqrt><mn>2</mn></msqrt></math>, then the area of quadrilateral is equal to :

&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>If</mi><mo>&nbsp;</mo><mfenced close="|" open="|"><mover accent="true"><mi>a</mi><mo>→</mo></mover></mfenced><mo>=</mo><mn>4</mn><mo>,</mo><mfenced close="|" open="|"><mover accent="true"><mi>b</mi><mo>→</mo></mover></mfenced><mo>=</mo><mn>2</mn><mo>&nbsp;</mo><mtext>  </mtext><mi>and</mi><mo>&nbsp;</mo><mtext>  </mtext><mfenced><mrow><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>,</mo><mover accent="true"><mi>b</mi><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mfrac><mi>π</mi><mn>6</mn></mfrac><mo>&nbsp;</mo><mi>t</mi><mi>h</mi><mi>e</mi><mi>n</mi></math>&nbsp;&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced close="|" open="|"><mrow><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>b</mi><mo>→</mo></mover></mrow></mfenced><mn>2</mn></msup><mo>=</mo></math>

&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>If</mi><mo>&nbsp;</mo><mn>2</mn><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>+</mo><mn>3</mn><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>+</mo><mn>4</mn><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>=</mo><mover accent="true"><mn>0</mn><mo>→</mo></mover><mo>⇒</mo></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>e</mi><mi>n</mi><mo>&nbsp;</mo><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>+</mo><mover accent="true"><mi>b</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>+</mo><mover accent="true"><mi>c</mi><mo>→</mo></mover><mo>×</mo><mover accent="true"><mi>a</mi><mo>→</mo></mover><mo>=</mo></math>

If&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><menclose notation="top"><mi mathvariant="normal">a</mi></menclose><mo>,</mo><mo>&nbsp;</mo><menclose notation="top"><mi mathvariant="normal">b</mi></menclose><mo>,</mo><mo>&nbsp;</mo><menclose notation="top"><mi mathvariant="normal">c</mi></menclose></math>&nbsp;are any three vectors then&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><menclose notation="top"><mi mathvariant="normal">a</mi></menclose><mo>×</mo><menclose notation="top"><mi mathvariant="normal">b</mi></menclose></mrow></mfenced><mo>×</mo><menclose notation="top"><mi mathvariant="normal">c</mi></menclose></math>&nbsp;is a vector

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>&nbsp;If&nbsp;</mtext><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mtext>&nbsp;where&nbsp;</mtext><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mtext>&nbsp;then the maximum&nbsp;&nbsp;value of&nbsp;</mtext><mo>(</mo><mn>4</mn><mi>a</mi><mo>−</mo><mn>3</mn><mi>b</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>5</mn><mi>b</mi><mo>−</mo><mn>4</mn><mi>c</mi><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>3</mn><mi>c</mi><mo>−</mo><mn>5</mn><mi>a</mi><msup><mo>)</mo><mn>2</mn></msup><mtext>&nbsp;is&nbsp;</mtext></math>

$\int (1 - \cos x) {cosec}^{2} xdx$ is equal to

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∫(1−cos⁡x)cosec2⁡xdx is equal to

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$\int (1 - \cos x) {cosec}^{2} xdx$ is equal to