Question

If cosecant A + cot A = 2/3 then find cosA​

Answer 1

Given:-

If cosecant A + cot A = 2/3

To find:-

CosA

Solution:-

CosecA=$\dfrac{1}{Sin \ A}$

CotA=$\dfrac{Cos \ A}{Sin \ A}$

[Converting the equation in terms of Cos A]

=$\dfrac{1}{Sin \ A}$ +$\dfrac{Cos \ A}{Sin \ A}$=$\dfrac{2}{3}$

∴3+Cos A=2 Sin A

[Squaring both sides we will get,]

9+9 Cos² A+ 18 Cos A=4 sin² A

9+9 Cos² A+18 Cos A=4-4 Cos² A

13 Cos² A+18 Cos A+5=0

[Solving quadratic equation by factoring]

13 Cos² A+13 Cos A+5 Cos A+5=0

13 Cos A (Cos A+1)+5(Cos A+1)=0

(13 Cos A+5)(Cos A+1)=0

Cos A=$\dfrac{-5}{13}$,-1