Advertisements
Advertisements
Question
Prove the following:
`cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
Advertisements
Solution
Let cos-1 x = α
Then, cos α = x, where 0 < α < π
Since, x > 0, 0 < α < `pi/2`
∴ sin α > 0, cos α > 0
Now, `tan^-1 (sqrt(1 - "x"^2)/"x") = tan^-1 (sqrt(1 - cos^2 alpha)/cos alpha)`
`= tan^-1 (sqrt(sin^2 alpha)/cos alpha)`
`= tan^-1 (tan alpha)`
`= alpha = cos^-1 "x"`
Hence, `cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
APPEARS IN
RELATED QUESTIONS
Find the principal solution of the following equation :
cot θ = `sqrt(3)`
Find the principal solution of the following equation:
tan θ = – 1
Find the general solution of the following equation :
cosθ = `sqrt(3)/(2)`
Find the general solution of the following equation:
tan θ = `(1)/(sqrt(3))`
Find the general solution of the following equation:
sec θ = `sqrt(2)`.
Find the general solution of the following equation:
tan `(2θ)/(3) = sqrt3`
Find the general solution of the following equation:
tan3θ = 3 tanθ.
Find the general solution of the following equation:
cosθ + sinθ = 1.
State whether the following equation have solution or not?
cos 2θ = – 1
Select the correct option from the given alternatives:
The principal solutions of equation sin θ = `- 1/2` are ______.
If `sqrt3 cos x − sin x = 1`, then general value of x is ______.
Select the correct option from the given alternatives:
In Δ ABC if ∠A = 45°, ∠B = 30°, then the ratio of its sides are
Select the correct option from the given alternatives:
In ΔABC if c2 + a2 – b2 = ac, then ∠B = ____.
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
`"cos"^-1 ("cos" (7pi)/6)` = _________.
The principal value of sin–1 `(- sqrt3/2)` is ______.
Select the correct option from the given alternatives:
If tan-1(2x) + tan-1(3x) = `pi/4`, then x = _____
Select the correct option from the given alternatives:
`2 "tan"^-1 (1/3) + "tan"^-1 (1/7) =` _____
If tan θ + tan 2θ + tan 3θ = tan θ.tan 2θ. tan 3θ, then the general value of the θ is ______.
Find the principal solutions of the following equation:
cot θ = 0
Find the general solutions of the following equation:
sin2 θ - cos2 θ = 1
Find the general solutions of the following equation:
sin θ - cos θ = 1
In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.
If 2 tan-1(cos x) = tan-1(2 cosec x), then find the value of x.
Show that `cos^-1 sqrt3/2 + 2 sin^-1 sqrt3/2 = (5pi)/6`.
Prove the following:
`cos^-1 "x" = pi + tan^-1 (sqrt(1 - "x"^2)/"x")`, if x < 0
If `tan^-1 "x" + tan^-1 "y" + tan^-1 "z" = pi/2,` then show that xy + yz + zx = 1
The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______
Find the principal solutions of sin x − 1 = 0
Find the principal solutions of cos 2ЁЭСе = 1
If cos–1x + cos–1y – cos–1z = 0, then show that x2 + y2 + z2 – 2xyz = 1
The value of tan 57°- tan 12°- tan 57° tan 12° is ______.
`[sin (tan^-1 3/4)]^2 + [sin(tan^-1 4/3)]^2 = ?`
If y = sin-1 `[(sqrt(1 + x) + sqrt(1 - x))/2]`, then `"dy"/"dx"` = ?
The values of x in `(0, pi/2)` satisfying the equation sin x cos x = `1/4` are ______.
If function
f(x) = `x - |x|/x, x < 0`
= `x + |x|/x, x > 0`
= 1, x = 0, then
The general solution of cot θ + tan θ = 2 is ______.
If `(tan 3 theta - 1)/(tan 3 theta + 1) = sqrt3`, then the general value of θ is ______.
The general solution of 4sin2 x = 1 is ______.
The general value of θ in the equation `2sqrt(3) cos theta = tan theta` is ______.
The general solution of cosec x = `-sqrt2` is ______
The equation 3sin2x + 10 cos x – 6 = 0 is satisfied, if ______.
If 2 tan–1(cos x) = tan–1(2 cosec x). then find the value of x.
General solution of the equation sin 2x – sin 4x + sin 6x = 0 is ______.
Which of the following equation has no solution?
