Question

42. If cosec A+ cotA=11/2
then cos A is equal to
A)21/22
B)15/16
C)117/125
(D) None of these
​

Answer 1

$\huge \underline{\bf{Solution}} : - \\ \red{ \bf{ \boxed{cos \: A = \frac{117}{125} }} } \\ \bf{Option \:(C )\:is \:correct.} \\ \\ \bf{ \underline{Find}} : - \\ \bf{cos \: A} \\ \\ A.T.Q.\\ \\ \: \bf{cosec \: A+ cot \: A = \frac{11}{2} } \\ \\ \bf{ \frac{1}{sin \: A} + \frac{cos \: A}{sin \: A} = \frac{11}{2} } \\ \\ \bold{\frac{1 + cos \: A}{sin \: A} = \frac{11 }{2} } \\ \\ \bf{1 + cos \: A = \frac{11}{2} sin \: A} \\ \bf{Squairing \: on \: both \: sides} \\ \\ \bf{ {(1 + cos \: A)}^{2} = \frac{121}{4} {sin}^{2} A} \\ \\ \bf{ {(1 + cos \: A)}^{2} = \frac{121}{4} (1 - {cos}^{2} A)} \\ \\ \bf{ {(1 + cos \: A)}^{2} = \frac{121}{4} (1 - cos \: A)(1 + cos \: A)} \\ \\ \bf{1 + cos \: A= \frac{121}{4} (1 - cos \: A)} \\ \\ \bf{ 1 + cos \: A = \frac{121}{4} - \frac{121}{4} cos \: A} \\ \\ \bf{cos \: A + \frac{121}{4} cos \: A = \frac{121}{4} - 1} \\ \\ \bf{cos \: A \bigg( \frac{125}{4} \bigg) = \frac{117}{4} } \\ \\ \bf{ \boxed{cos \: A = \frac{117}{125} }} \\ \\ \bf{ \underline{formula \: used}} : - \\ \\ \bf{ \star \: (a - b)(a + b) = {a}^{2} - {b}^{2} } \\ \\ \bf{ \star \: \frac{1}{sin \theta} = cosec \theta }\\ \\ \bf{\star \: \frac{cos \theta}{sin \theta} = cot \theta}$