The minimum value of, t a n 2 β s e c 4 α + t a n 2 α s e c 4 β , α , β  = 2 K π , K ∈ I , is

b ( a + 1 ) 2 + a ( b + 1 ) 2 = b a 2 + b 1 + a b 2 + a 1 + 2 ( b a + a b ) ≥ 4 [ b a 2 ⋅ b 1 ⋅ a b 2 ⋅ a 1 ] 4 1 + 4 ( b a ⋅ a b ) 2 1 ≥ 8 Where a = tan 2 α , b = tan 2 β

Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

MathematicsTrigonometric Ratios & IdentitiesMaximum & Minimum ValuesMedium2 minQB

MathematicsMediuminteger

The minimum value of, $\frac{s e c ^{4} α}{t a n ^{2} β} + \frac{s e c ^{4} β}{t a n ^{2} α}, α, β \neq = \frac{K π}{2}, K \in I$ , is

Answer:

Solution:

$\frac{( a + 1 ) ^{2}}{b} + \frac{( b + 1 ) ^{2}}{a} = \frac{a ^{2}}{b} + \frac{1}{b} + \frac{b ^{2}}{a} + \frac{1}{a} + 2 (\frac{a}{b} + \frac{b}{a}) \geq 4 [\frac{a ^{2}}{b} \cdot \frac{1}{b} \cdot \frac{b ^{2}}{a} \cdot \frac{1}{a}]^{\frac{1}{4}} + 4 (\frac{a}{b} \cdot \frac{b}{a})^{\frac{1}{2}} \geq 8$

Where $a = tan^{2} α, b = tan^{2} β$

Stream:JEESubject:MathematicsTopic:Trigonometric Ratios & IdentitiesSubtopic:Maximum & Minimum Values

⏱ 2mℹ️ Source: QB

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