Advertisements
Advertisements
प्रश्न
Prove that `1/2 "cos"^(-1) ((1-"x")/(1+"x")) = "tan"^-1 sqrt"x"`
उत्तर
L.H.S. = `1/2"cos"^-1((1-"x")/(1+"x"))`
Put x = tan2θ ⇒ tan θ = `sqrt"x" => theta = "tan"^-1 sqrt"x"`
`= 1/2 "cos"^-1 ((1 - "tan"^2theta)/(1 + "tan"^2theta)) = 1/2 "cos"^-1 ("cos"2theta)`
`= 1/2 xx 2theta = theta = "tan"^-1 sqrt"x" = "R.H.S"`
APPEARS IN
संबंधित प्रश्न
Find the values of p and q for which
f(x) = `{((1-sin^3x)/(3cos^2x),`
is continuous at x = π/2.
Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
Find all points of discontinuity of f, where f is defined by `f(x) = {(|x|+3, if x<= -3),(-2x, if -3 < x < 3),(6x + 2, if x >= 3):}`
Find all points of discontinuity of f, where f is defined by `f(x) = {(|x|/x , if x != 0),(0, if x = 0):}`
Find all points of discontinuity of f, where f is defined by `f (x) = {(x/|x|, if x<0),(-1, if x >= 0):}`
Find all points of discontinuity of f, where f is defined by `f (x) = {(x+1, if x>=1),(x^2+1, if x < 1):}`
Find all points of discontinuity of f, where f is defined by `f(x) = {(x^3 - 3, if x <= 2),(x^2 + 1, if x > 2):}`
Is the function defined by `f(x) = {(x+5, if x <= 1),(x -5, if x > 1):}` a continuous function?
Show that the function defined by g(x) = x = [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.
Find the points of discontinuity of f, where `f (x) = {(sinx/x, if x<0),(x + 1, if x >= 0):}`
Find the relationship between 'a' and 'b' so that the function 'f' defined by
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}- 2 , & \text{ if }& x \leq - 1 \\ 2x , & \text{ if } & - 1 < x < 1 \\ 2 , & \text{ if } & x \geq 1\end{cases}\]
Find the point of discontinuity, if any, of the following function: \[f\left( x \right) = \begin{cases}\sin x - \cos x , & \text{ if } x \neq 0 \\ - 1 , & \text{ if } x = 0\end{cases}\]
Let f (x) `= (1 - "tan x")/(4"x" - pi), "x" ne pi/4, "x" in (0, pi/2).` If f(x) is continuous in `(0, pi/2), "then f"(pi/4) =` ____________.
If f(x) `= sqrt(4 + "x" - 2)/"x", "x" ne 0` be continuous at x = 0, then f(0) = ____________.
`lim_("x"-> 0) sqrt(1/2 (1 - "cos" 2"x"))/"x"` is equal to
The point of discountinuity of the function `f(x) = {{:(2x + 3",", x ≤ 2),(2x - 3",", x > 2):}` is are
`f(x) = {{:(x^3 - 3",", if x < 2),(x^2 + 1",", if x > 2):}` has how many point of discontinuity
Sin |x| is a continuous function for
Let α ∈ R be such that the function
f(x) = `{{:((cos^-1(1 - {x}^2)sin^-1(1 - {x}))/({x} - {x}^3)",", x ≠ 0),(α",", x = 0):}`
is continuous at x = 0, where {x} = x – [x], [x] is the greatest integer less than or equal to x.
If the function f defined as f(x) = `1/x - (k - 1)/(e^(2x) - 1)` x ≠ 0, is continuous at x = 0, then the ordered pair (k, f(0)) is equal to ______.
Find the value of k for which the function f given as
f(x) =`{{:((1 - cosx)/(2x^2)",", if x ≠ 0),( k",", if x = 0 ):}`
is continuous at x = 0.
If f(x) = `{{:((kx)/|x|"," if x < 0),( 3"," if x ≥ 0):}` is continuous at x = 0, then the value of k is ______.
The graph of the function f is shown below.
Of the following options, at what values of x is the function f NOT differentiable?
