Question

cosecA+cotA=2/3then cosA=?​

Answer 1

Given that

$cosec A + cot A = \frac{2}{3} \\\\( putting \ cosecA = 1/Sin A \\\ and \ cot A = cos A / sin A \ and \ taking \ LCM )\\\\\frac{1}{sin A} + \frac{cos A}{sin A} = \frac{2}{3}\\\\ \frac{1+cos A}{sin A} = \frac{2}{3} \\\\( cross \ multiplying )\\\\3 + 3 cos A = 2 sin A\\\\( squaring both sides )\\\\ ( 3 + 3 cos A )^2 = ( 2 sin A ) ^2\\\\9 + 9 cos^2A + 18 cos A = 4 sin^2 A\\\\( putting \ sin^2A = 1 - cos^2 A)\\\\9 cos^2 A + 18 cos A + 9 = 4 ( 1 - cos^2 A)$

$9 cos^2 A + 18 cos A + 9 = 4 - 4 cos^2 A\\\\9 cos^2A + 4 cos^2A + 18 cos A + 9 - 4 = 0\\\\13 cos^2 A + 18 cos A + 5 = 0\\\\(let, us\ take \ cos A = x )\\\\13 x^2 + 18 x + 5 = 0\\\\13 x ^2 + 13 x +5x +5 =0\\\\13 x ( x + 1 ) + 5 ( x + 1 ) = 0\\\\(13 x +5) ( x + 1 ) = 0\\\\x = -5/13 \\\ and \\x = -1$

HENCE,

there are two possible values of cos A

cos A = -5 /13

or, cos A = -1

( these both values are worthy here because range of values of cos A is greater than or equal to -1 and less than or equal to (positive) 1.