Determine-the-minimum-value-sec-4-tan-2-sec-4-tan-2-over-all-kpi-2-k-Z-

Question Number 177793 by cortano1 last updated on 09/Oct/22

Determine the minimum value

\frac{\sec^{4} α}{\tan^{2} β} + \frac{\sec^{4} β}{\tan^{2} α}

over all α, β \neq \frac{k π}{2}, k ε Z

Answered by mahdipoor last updated on 09/Oct/22

g e t {t a n}^{2} x = z \Rightarrow

{s e c}^{4} x = {(\frac{1}{{c o s}^{2} x})}^{2} = {(1 + {t a n}^{2} x)}^{2} = {(1 + z)}^{2}

\Rightarrow\Rightarrow

\Rightarrow {t a n}^{2} β = m \Rightarrow {s e c}^{4} β = {(1 + m)}^{2}

\Rightarrow {t a n}^{2} α = n \Rightarrow {s e c}^{4} α = {(1 + n)}^{2}

\Rightarrow\Rightarrow

f (α, β) = f (m, n) = \frac{{(1 + m)}^{2}}{n} + \frac{{(1 + n)}^{2}}{m} m, n \neq 0

{\begin{cases} \frac{\partial f}{\partial m} = \frac{2 (1 + m)}{n} - \frac{{(1 + n)}^{2}}{m^{2}} = 0 \\ \frac{\partial f}{\partial n} = - \frac{{(1 + m)}^{2}}{n^{2}} + \frac{2 (1 + n)}{m} = 0 \end{cases}

\Rightarrow m = n = 1

m i n f = f (1, 1) = 8