Question

if cosec a + cot a = k
then prove that cos a = (k²-1)/(k²+1)​

Answer 1

Answer:

Step-by-step explanation:

cosec β + cot β = k

⇒1/sin β + cos β/sin β = k

⇒(1 + cos β) / sin β = k

⇒(1 + cos β) / √(1 - cos² β) = k

⇒(1 + cos β) / √(1 + cos β)(1 - cos β)= k

⇒√[(1 + cos β) / (1 - cos β)] = k

⇒(1 + cos β) / (1 - cos β) = k²/1

⇒[(1 + cos β) - (1 - cos β) ]/[(1 - cos β) + (1 + cos β) ] = (k²-1)/(k²+1)

(using the formula if \frac{a}{b} =  \frac{c}{d}

then  \frac{a-b}{a+b} =  \frac{c-d}{c+d} )

⇒(2 cos β)/ 2 = (k²-1)/(k²+1)

⇒cos β = (k²-1)/(k²+1)