Question

If $4{\sin }^{2}A-13\sin A+3=0.$ Find value of $32{\cos }^{2}A+{\cot }^{2}A$

Answer 1

$4si{n}^{2}A-12sinA-sinA+3=04sinA(sinA-3)-1(sinA-3)=0(4sinA-1)(sinA-3)=0\therefore sinA=\left(\frac{1}{4}\right)or3butsinA\ne 3\therefore sinA=\left(\frac{1}{4}\right)soco{s}^{2}A=1-si{n}^{2}A=1-\left(\frac{1}{16}\right)=\left(\frac{15}{16}\right)\therefore 32co{s}^{2}A+co{t}^{2}A=32\times \left(\frac{15}{16}\right)+\left(\frac{\left(\frac{15}{16}\right)}{\left(\frac{1}{16}\right)}\right)=30+15=42$