The minimum value of 4 cosec2 x-32x-2 is

4 cosec^2 x+3^2x+2 =4(1+^2x)+3(1+tan^2x)+2 =4+4^2x+3+3tan^2x+2 =9+4^2x+3tan^2x We know that, a+b≥2√(ab) if a,b gt;0 Let nbsp;a=4^2x, b=3tan^2x 4^2x+3ta ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Range of Trigonometric Expressions

The minimum v...

Question

The minimum value of $4 {cosec}^{2} x + 3 {sec}^{2} x + 2$ is

9 + 2 \sqrt{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

9 - 2 \sqrt{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

9 + 4 \sqrt{3}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

9 - 4 \sqrt{3}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $9 + 4 \sqrt{3}$
$4 {cosec}^{2} x + 3 {sec}^{2} x + 2 = 4 (1 + {cot}^{2} x) + 3 (1 + {tan}^{2} x) + 2 = 4 + 4 {cot}^{2} x + 3 + 3 {tan}^{2} x + 2 = 9 + 4 {cot}^{2} x + 3 {tan}^{2} x$

We know that, $a + b \geq 2 \sqrt{a b}$ if $a, b > 0$

Let $a = 4 {cot}^{2} x, b = 3 {tan}^{2} x$
$4 {cot}^{2} x + 3 {tan}^{2} x \geq 4 \sqrt{3} \cdot \sqrt{{cot}^{2} x \cdot {tan}^{2} x} \Rightarrow 9 + 4 {cot}^{2} x + 3 {tan}^{2} x \geq 9 + 4 \sqrt{3}$

Suggest Corrections

Similar questions

Q. The value of

{sec}^{2} ({tan}^{- 1} 2) + {cosec}^{2} ({cot}^{- 1} 3) =

____

Q. Evaluate :

\int \frac{{sec}^{2} x}{{cosec}^{2} x} d x

Q. The value of sec²(tan^-12) + cosec² (cot^-1 3) is ____________________.

Q. The value of

{c o s e c}^{2} 51^{0} - {c o t}^{2} 51^{0}

Q. If

sin θ + c o s e c θ = 2

then find the value of

{sin}^{2} θ + {c o s e c}^{2} θ

Join BYJU'S Learning Program

The minimum value of 4 cosec2 x+3sec2x+2 is

The correct option is C 9+4√34 cosec2 x+3sec2x+2=4(1+cot2x)+3(1+tan2x)+2=4+4cot2x+3+3tan2x+2=9+4cot2x+3tan2x We know that, a+b≥2√ab if a,b>0 Let a=4cot2x, b=3tan2x 4cot2x+3tan2x≥4√3⋅√cot2x⋅tan2x⇒9+4cot2x+3tan2x≥9+4√3

The minimum value of $4 {cosec}^{2} x + 3 {sec}^{2} x + 2$ is