Question

if cosecA + cotA = m say that m2-1 by m2+1 = cos A​

Answer 1

GIVEN:

• Cosec A + Cot A = m

TO PROVE:

• (m² - 1) / (m² + 1) = cos A

FORMULAE:

★ 1 = Cosec² A - Cot² A

★ A² - B² = (A + B)(A - B) [(Cosec² A - Cot² A) = (Cosec A + Cot A)(Cosec A - Cot A) ]

★ Cot A = Cos A / Sin A

★ Cosec A = 1 / Sin A

PROOF:

m² - 1 = (Cosec A + Cot A)² - 1

m² - 1 = (Cosec A + Cot A)² - (Cosec² A - Cot² A)

Take Cosec A + Cot A as common

m² - 1 = (Cosec A + Cot A)(Cosec A + Cot A - Cosec A + Cot A )

m² - 1 = (Cosec A + Cot A)(2 Cot A )

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m² + 1 = (Cosec A + Cot A)² + 1

m² + 1 = (Cosec A + Cot A)² + (Cosec² A - Cot² A)

Take Cosec A + Cot A as common

m² + 1 = (Cosec A + Cot A)(Cosec A + Cot A + Cosec A - Cot A )

m² + 1 = (Cosec A + Cot A)(2 Cosec A)

${\dfrac{(m^2 - 1)}{ (m^2 + 1)}= \dfrac{(Cosec\ A + Cot\ A)(2\ Cot\ A )}{(Cosec\ A + Cot\ A)(2\ Cosec\ A )}}$

$\dfrac{(m^2 - 1)}{ (m^2 + 1)}= \dfrac{Cot\ A }{Cosec\ A }$

$\dfrac{(m^2 - 1)}{ (m^2+1)}=\dfrac{\dfrac{Cos \ A}{Sin \ A} }{\dfrac{1}{Sin \ A}}$

$\large{\bold{\gray{\bigstar\dfrac{(m^2 - 1)}{ (m^2 + 1)}= Cos \ A}}}$