Question

Given that cosec A+cotA=x.Prove that COSA=x²-1÷x²+1

Answer 1

$use \: \: \boxed{sin^{2}A = 1 - cos^{2}A }$

Step-by-step explanation:

$Given, \: cosec A + cot A = x \\ \frac{1}{sin A} + \frac{cos A}{sin A} = x \\ \frac{1 + cos A}{sin A} = x \\ 1 + cos A = x.sin A \\ On \: \: square \: \: off \: \: both \: \: sides, \\(1 + cos A)^{2} =( x.sin A)^{2} \\ 1 + cos^{2} A + 2cosA = {x}^{2} .sin^{2}A \\ 1 + cos^{2} A + 2cosA = {x}^{2} .(1 - cos^{2}A) \\ 1 + cos^{2} A + 2cosA = {x}^{2} - {x}^{2} cos^{2}A \\ 1 + cos^{2} A + 2cosA - {x}^{2} + {x}^{2} cos^{2}A = 0 \\ ( {x}^{2} + 1 )cos^{2} A + 2cosA - ( {x}^{2} - 1 ) = 0 \\ ( {x}^{2} + 1 )cos^{2} A + ( {x}^{2} + 1)cosA - ( {x}^{2} - 1 )cosA - ( {x}^{2} - 1 ) = 0 \\ ( {x}^{2} + 1 )cosA(cosA + 1) - ( {x}^{2} - 1)(cosA + 1) = 0 \\ (cosA + 1) \{ ( {x}^{2} + 1 )cosA - ( {x}^{2} - 1 )\} = 0 \\ ( {x}^{2} + 1 )cosA - ( {x}^{2} - 1 ) = 0 \\ ( {x}^{2} + 1 )cosA = ( {x}^{2} - 1 ) \\ cosA = \frac{ {x}^{2} - 1 }{ {x}^{2} + 1 } .$