Question

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Answer 1

Question

$( \bf \: \csc(A) - \sin(A) )( \sec(A) - \cos(A) )( \tan(A) + \cot(A) ) = 1$

Solution:-

$( \bf \: \csc(A) - \sin(A) )( \sec(A) - \cos(A) )( \tan(A) + \cot(A) ) = 1$

By Using this trigonometry identities

$\to \: \csc(A) = \frac{1}{ \sin(A) }$

$\to \sec(A) = \frac{1}{ \cos(A) }$

$\to \: \tan(A) = \frac{ \sin(A) }{ \cos(A) }$

$\to \: \cot(A) = \frac{ \cos(A) }{ \sin(A) }$

We get

$( \bf \: \frac{1}{ \sin(A) } - \sin(A) )( \frac{1}{ \cos(A) } - \cos(A) )( \frac{ \sin(A) }{ \cos(A) } + \frac{ \cos(A) }{ \sin(A) } )$

Taking lcm we get

$\bf \: ( \frac{1 - \sin {}^{2} (A) }{ \sin(A) } )( \frac{1 - \cos {}^{2} (A) }{ \cos(A) } )( \frac{ \sin {}^{2} (A) + \cos {}^{2} (A) }{ \sin(A) \cos(A) } )$

Use this trigonometry identities

=> sin²(A) + cos²(A) = 1

=> sin²(A) = 1 - cos²(A)

=> cos²(A) = 1 - sin²(A)

$\bf \: ( \frac{ \cos {}^{2} (A) }{ \sin(A) } )( \frac{ \sin {}^{2} (A) }{ \cos {}^{} (A) } )( \frac{1}{ \sin(A) \cos(A) } )$

$\bf \: ( \frac{ \not\cos {}^{ \not2} (A) }{ \sin(A) } )( \frac{ \sin {}^{2} (A) }{ \not{\cos }{}^{} (A) } )( \frac{1}{ \sin(A) \not\cos(A) } )$

$\bf \: \frac{ \not \sin {}^{2} (A) }{ \not\sin(A) } \times \frac{1}{ \not\sin(A) }$

$\implies \: 1$

$\bf \: { \red{hence \: proved}}$