Home
Class 12
MATHS
Using the relation 2(1-cosx)<x^2,x!=0 or...

Using the relation `2(1-cosx)

Text Solution

Verified by Experts

Let `f (x)= sin( tan x ) - x `
`f' (x) = cos ( tan x )* sec^(2) x - 1 `
`" " = cos ( tan x ) ( 1 + tan ^(2) x) -1 `
`= tan ^(2)x {cos ( tan x ) } + cos (tan x ) - 1 `
`gt tan ^(2) x cos (tan x ) - ( tan ^(2)x)/( 2)`
`[{:((because 2 (1- cosx ) lt x ^(2), x ne 0 rArr cos x gt 1- ( x^(2))/(2))),(rArr " " cos (tanx ) gt 1 - ( tan ^(2) x)/(2)):}]`
`f'(x) gt tan ^(2)x [ cos (tan x ) - cos ( pi //3) ] gt 0`
`rArr f(x)` is increasing function, for all ` x in [ 0, pi//4]`
`rArr " " sin (tan x ) ge x `
Promotional Banner

Similar Questions

Explore conceptually related problems

Find the integrals of the functions (1-cosx)/(1+cosx)

Find the derivative of (cosx)/(1+sinx)

intsinx/(1-cosx)dx

Find the range of the function (1)/(2cosx-1) .

Find the integrals of the functions (cosx)/(1+cosx)

Show that range of the function (1)/(2cosx-1)" is "(-oo,-(1)/(3)]cup[1,oo)

Find the range of f(x)=1/(2cosx-1)

Find the imtegrals of the functions (cosx-sinx)/(1+sin2x)

Integrate the functions (2cosx-3sinx)/(6cosx+4sinx)