Advertisements
Advertisements
Question
The number of real solutions of the equation `sqrt(1 + cos 2x) = sqrt(2) cos^-1 (cos x)` in `[pi/2, pi]` is ______.
Options
0
1
2
Infinite
Advertisements
Solution
The number of real solutions of the equation `sqrt(1 + cos 2x) = sqrt(2) cos^-1 (cos x)` in `[pi/2, pi]` is 0.
Explanation:
We have `sqrt(1 + cos 2x) = sqrt(2) cos^-1 (cos x)`
⇒ `sqrt(2 cos^2x) = sqrt(2)x` ...`[because cos^-1 (cos x) = x]`
⇒ `sqrt(2) cos x = sqrt(2)x`
⇒ cos x = x
∴ There are no solution for given equation.
APPEARS IN
RELATED QUESTIONS
If `sin (sin^(−1) 1/5+cos^(−1) x)=1`, then find the value of x.
Prove that: `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4`
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
Write the function in the simplest form: `tan^(-1) 1/(sqrt(x^2 - 1)), |x| > 1`
Write the following function in the simplest form:
`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt(3) < x < a/sqrt(3)`
Find the value of the following:
`tan 1/2 [sin^(-1) (2x)/(1 + x^2) + cos^(-1) (1 - y^2)/(1 + y^2)], |x| < 1, y > 0 and xy < 1`
Prove that `sin^(-1) 8/17 + sin^(-1) 3/5 = tan^(-1) 77/36`.
Prove `tan^(-1) 1/5 + tan^(-1) (1/7) + tan^(-1) 1/3 + tan^(-1) 1/8 = pi/4`
Prove that `tan {pi/4 + 1/2 cos^(-1) a/b} + tan {pi/4 - 1/2 cos^(-1) a/b} = (2b)/a`
Prove that
\[2 \tan^{- 1} \left( \frac{1}{5} \right) + \sec^{- 1} \left( \frac{5\sqrt{2}}{7} \right) + 2 \tan^{- 1} \left( \frac{1}{8} \right) = \frac{\pi}{4}\] .
Solve for x : \[\cos \left( \tan^{- 1} x \right) = \sin \left( \cot^{- 1} \frac{3}{4} \right)\] .
Solve: tan-1 4 x + tan-1 6x `= π/(4)`.
Find the value of the expression in terms of x, with the help of a reference triangle
sin (cos–1(1 – x))
Prove that `tan^-1 2/11 + tan^-1 7/24 = tan^-1 1/2`
Solve: `tan^-1x = cos^-1 (1 - "a"^2)/(1 + "a"^2) - cos^-1 (1 - "b"^2)/(1 + "b"^2), "a" > 0, "b" > 0`
Choose the correct alternative:
The equation tan–1x – cot–1x = `tan^-1 (1/sqrt(3))` has
Choose the correct alternative:
If `sin^-1x + cot^-1 (1/2) = pi/2`, then x is equal to
Choose the correct alternative:
sin(tan–1x), |x| < 1 is equal to
Evaluate: `tan^-1 sqrt(3) - sec^-1(-2)`.
Solve the equation `sin^-1 6x + sin^-1 6sqrt(3)x = - pi/2`
Prove that `tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/((1 + x^2) - sqrt(1 - x^2))) = pi/2 + 1/2 cos^-1x^2`
If |x| ≤ 1, then `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` is equal to ______.
If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then ______ < y < ______.
The maximum value of sinx + cosx is ____________.
The value of `"tan"^ -1 (3/4) + "tan"^-1 (1/7)` is ____________.
`"cot" (pi/4 - 2 "cot"^-1 3) =` ____________.
The domain of the function defind by f(x) `= "sin"^-1 sqrt("x" - 1)` is ____________.
If sin `("sin"^-1 1/5 + "cos"^-1 "x") = 1,` then the value of x is ____________.
Simplest form of `tan^-1 ((sqrt(1 + cos "x") + sqrt(1 - cos "x"))/(sqrt(1 + cos "x") - sqrt(1 - cos "x")))`, `π < "x" < (3π)/2` is:
The value of `"cos"^-1 ("cos" ((33pi)/5))` is ____________.
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
`"tan"^-1 1 + "cos"^-1 ((-1)/2) + "sin"^-1 ((-1)/2)`
`"sin"^-1 (1/sqrt2)`
`"cos"^-1 (1/2)`
If `3 "sin"^-1 ((2"x")/(1 + "x"^2)) - 4 "cos"^-1 ((1 - "x"^2)/(1 + "x"^2)) + 2 "tan"^-1 ((2"x")/(1 - "x"^2)) = pi/3` then x is equal to ____________.
Find the value of `cos^-1 (1/2) + 2sin^-1 (1/2) ->`:-
`50tan(3tan^-1(1/2) + 2cos^-1(1/sqrt(5))) + 4sqrt(2) tan(1/2tan^-1(2sqrt(2)))` is equal to ______.
Find the value of `tan^-1 [2 cos (2 sin^-1 1/2)] + tan^-1 1`.
The value of cosec `[sin^-1((-1)/2)] - sec[cos^-1((-1)/2)]` is equal to ______.
