$frac{4}{1+ an ^{2} alpha}+frac{1}{1+cot ^{2} alpha}+3 sin ^{2} alpha$ का मूल्य

$frac{4}{1+ an ^{2} alpha}+frac{1}{1+cot ^{2} alpha}+3 sin ^{2} alpha$ $=frac{4}{sec ^{2} alpha}+frac{1}{operatorname{cosec}^{2} alpha}+3 sin ^{2} alpha$ $=4 cos ^{2} alpha+sin ^{2} alpha+3 sin ^{2} alpha$ $=4left(cos ^{2} alpha+sin ^{2} alpha ight)=4$

विकल्प:

मान लीजिए α = 45°

$frac{4}{1+ an ^{2} alpha}+frac{1}{1+cot ^{2} alpha}+3 sin ^{2} alpha$

tan 45° = cot 45° = 1

$=frac{4}{2}+frac{1}{2}+3 imesleft(frac{1}{sqrt{2}} ight)^{2}=frac{8}{2}=4$