Question

18. (cotA – cosecA)2 = 1 – cosA/1 + cosA

Answer 1

♦ Refer to above attachment ↑

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→Trigonometric Identities:

sin2A + cos2A = 1

tan2A + 1 = sec2A

cot2A + 1 = cosec2A

→Relation Between Trigonometric Identities:

• tanA = sinA/cosA

• cotA = cosA/sinA

• cosecA = 1/sinA

• secA = 1/cosA

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Answer 2

Step-by-step explanation:

We Have :-

$( \cot(a) - \csc(a) )^{2} = \dfrac{1 - \cos(a) }{1 + \cos(a) }$

Identities Used :-

$\cot(a) = \dfrac{ \cos(a) }{ \sin(a) } \\ \\ \\ \csc(a) = \dfrac{1}{ \sin(a) } \\ \\ \\ a^{2} - b^{2} = (a + b)(a - b) \\ \\ \\ \sin^{2} (a) = 1 - \cos^{2} (a)$

Solution :-

$( \cot(a) - \csc(a) )^{2} = \dfrac{1 - \cos(a) }{1 + \cos(a) } \\ \\ \\ ( \cot(a) - \csc(a) )^{2} \\ \\ \\ ( \dfrac{ \cos(a) }{ \sin(a) } - \dfrac{1}{ \sin^{2} (a) } )^{2} \\ \\ \\ ( \dfrac{ \cos(a) - 1 }{ \sin(a) } )^{2} \\ \\ \\ \dfrac{ (\cos(a) - 1)^{2} }{ \sin ^{2} (a) } \\ \\ \\ \dfrac{ (\cos(a) - 1)( \cos(a) - 1)}{ 1 - \cos^{2} (a) } \\ \\ \\ \dfrac{ (\cos(a) - 1)( \cos(a) - 1)}{ (1 - \cos (a) )(1 - \cos(a)) } \\ \\ \\ \dfrac{1 - \cos(a) }{1 + \cos(a) }$