Minimum value of $5 an ^2 \alpha+\frac{9}{ an ^2 \alpha}+4 \sec ^2 \alpha$ is

Question

Minimum value of $5 	an ^2 \alpha+\frac{9}{	an ^2 \alpha}+4 \sec ^2 \alpha$ is

Aicharya · Accepted Answer

$5 	an ^2 \alpha+\frac{9}{	an ^2 \alpha}+4 \sec ^2 \alpha$ $=5 	an ^2 \alpha+9 \cot ^2 \alpha+4\left(1+	an ^2 \alphaight)$ $\left[\because \sec ^2 x=1+	an ^2 xight]$ $=9 	an ^2 \alpha+9 \cot ^2 \alpha+4$ Since, we know that $A M \geq G M$ $\frac{9 	an ^2 \alpha+9 \cot ^2 \alpha}{2} \geq \sqrt{9 	an ^2 \alpha \cdot 9 \cot ^2 \alpha}$ $9 	an ^2 \alpha+9 \cot ^2 \alpha \geq 9 \cdot 2 \cdot 	an \alpha \cdot \cot \alpha$ $9 	an ^2 \alpha+9 \cot ^2 \alpha \geq 18$ $	herefore \quad 9 	an ^2 \alpha+9 \cot ^2 \alpha+4 \geq 18+4=22$ $	herefore$ Hence, minimum values $=22$

Minimum value of $5 \tan ^2 \alpha+\frac{9}{\tan ^2 \alpha}+4 \sec ^2 \alpha$ is

Solution