Start on the left side.

Write <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mstyle></math> in sines and cosines using the quotient identity.

Apply the reciprocal identity to <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mfrac><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> .

Multiply the numerator and denominator of the complex fraction by <math><mstyle displaystyle="true"><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mfrac><mrow><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>-</mo><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mfrac><mrow><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mstyle></math> .

Combine.

Apply the distributive property.

Simplify by cancelling.

Cancel the common factor of <math><mstyle displaystyle="true"><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of <math><mstyle displaystyle="true"><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Simplify the numerator.

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> out of <math><mstyle displaystyle="true"><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mo>-</mo><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mstyle></math> .

Multiply by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> out of <math><mstyle displaystyle="true"><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mo>-</mo><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> out of <math><mstyle displaystyle="true"><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>1</mn><mo>-</mo><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> in a factored form.

Rewrite <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Since both terms are perfect squares, factor using the difference of squares formula, <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>)</mo></mrow></mstyle></math> where <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mstyle></math> .

Simplify the denominator.

One to any power is one.

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>1</mn><mo>-</mo><msup><mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> in a factored form.

Rewrite <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Since both terms are perfect squares, factor using the difference of squares formula, <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>)</mo></mrow></mstyle></math> where <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>1</mn><mo>+</mo><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>1</mn><mo>-</mo><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mstyle></math> .

Because the two sides have been shown to be equivalent, the equation is an identity.

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