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JEE Advanced

Exam: JEE Advanced
Subject: Mathematics
Chapter: Trigonometry

JEE Advanced · Mathematics · Trigonometry

What is the minimum value of ( ^4 α)/( ^2 β)+( ^4 β)/( ^2 α) for α,...

What is the minimum value of

\frac{\sec ^4 \alpha}{\tan ^2 \beta}+\frac{\sec ^4 \beta}{\tan ^2 \alpha}

for

\alpha, B \neq \frac{k \pi}{2}, k \in Z

?

Options

A $2$
B $4$
C $8$
D $16$

Correct answer

C. $8$

Step-by-step solution

\tan ^2 \alpha \rightarrow \mathrm{a}, \tan ^2 \mathrm{~B} \rightarrow \mathrm{b}\frac{(a+1)^2}{b}+\frac{(b+1)^2}{a}=\left[\frac{a^2}{b}+\frac{1}{b}+\frac{b^2}{a}+\frac{1}{a}\right]+2\left(\frac{a}{b}+\frac{b}{a}\right) \geq 4\left[\frac{a^2}{b} \cdot \frac{1}{b} \cdot \frac{b^2}{a} \cdot \frac{1}{a}\right]^{1 / 4}+2 \cdot 2\left[\frac{a}{b} \cdot \frac{b}{a}\right]^{1 / 2}=8

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