Using intermediate value theorem, prove that there exists a number `x` such that `x^(2005)+1/(1+sin^2x)=2005.`

Using Lagrange's mean value theorem prove that, e^(x) gt 1 +x

Using Largrange's mean value theorem, prove that, tan^(-1)x lt x

Use the mean value theorem to prove e^(x)ge1+xAA x(inR)

Using Lagrange's mean value theorem prove that, log (1+x) lt x ("when " x gt 0)

Using the remainder theorem proved that (2x - 1) is a factor of the polynomial (4x^(3)-4x^(2)-x+1) .

Prove that , (sin2x)/(1+cos2x)=tanx

Let f(x)=(1-x)^(2) sin^(2) x+x^(2) for all x in RR , and let g(x)=int_(1)^(x) ((2(t-1))/(t+1)-log t)f(t) dt for all x in (1, oo) . Consider the statements : P : there exists some x in RR such that f(x)+2x=2(1+x^(2)) Q : There exists some x in RR such that 2f(x)+1=2x(1+x) then-

if sin^4x+sin^2x=1 ,then prove that cot^4x+cot^2x =1.

Using the remainder theorem, examine whether (2x + 1) is a multiple of the polynomial p(x)=4x^(3)+4x^(2)-x-1 or not.

If sec theta+tan theta = x , prove that sin theta= (x^2-1)/(x^2+1)