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Inverse Trigonometric Functions Test - 68

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Inverse Trigonometric Functions Test - 68
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  • Question 1
    1 / -0
    tan1(1x21x2)+sin1(x2+1x21)tan^{-1}(1-x^{2}-\frac{1}{x^{2}})+sin^{-1}(x^{2}+\frac{1}{x^{2}}-1) is equal to 
    Solution

  • Question 2
    1 / -0
    Evaluate : sin (12cos145)\sin \left(\dfrac {1}{2}\cos^{-1}\dfrac {4}{5}\right) 
    Solution

  • Question 3
    1 / -0
    If cot11x+cot11y+cot11z=π2cot^{1}\frac{1}{x}+cot^{-1}\frac{1}{y}+cot^{-1 }\frac{1}{z}=\frac{\pi }{2} then.....
    Solution

  • Question 4
    1 / -0
    If α=2tan1(322)+sin1(162),β=cot1(32)+18sec1(2)\alpha =2\tan^{-1}(\sqrt{3-2\sqrt{2}})+\sin^{-1}\left(\dfrac{1}{\sqrt{6}-\sqrt{2}}\right), \beta =\cot^{-1}(\sqrt{3}-2)+\dfrac{1}{8}\sec^{-1}(-2) & γ=tan112+cos113\gamma =\tan^{-1}\dfrac{1}{\sqrt{2}}+\cos^{-1}\dfrac{1}{\sqrt{3}}, then?
  • Question 5
    1 / -0
    If cos1x>sin1x\cos^{-1} x > \sin^{-1} x, then find the range of x
    Solution
    We have cos1x>sin1x{\cos}^{-1}{x}>{\sin}^{-1}{x}

    π2sin1x>sin1x\Rightarrow\,\dfrac{\pi}{2}-{\sin}^{-1}{x}>{\sin}^{-1}{x}

    π2>2sin1x\Rightarrow\,\dfrac{\pi}{2}>2{\sin}^{-1}{x}

    π4>sin1x\Rightarrow\,\dfrac{\pi}{4}>{\sin}^{-1}{x}

    sin1x<π4\Rightarrow\,{\sin}^{-1}{x}<\dfrac{\pi}{4}    .......(1)(1)

    But π2sin1x<π2-\dfrac{\pi}{2}\le {\sin}^{-1}{x}<\dfrac{\pi}{2}      ........(2)(2)

    From (1)(1) and (2)(2) we have

    π2sin1x<π4-\dfrac{\pi}{2}\le {\sin}^{-1}{x}<\dfrac{\pi}{4}

    sin(π2)x<sinπ4\Rightarrow\,\sin{\left(-\dfrac{\pi}{2}\right)}\le x<\sin{\dfrac{\pi}{4}}

    1x<12\Rightarrow\,-1\le x<\dfrac{1}{\sqrt{2}}

  • Question 6
    1 / -0
    4sin1x+cos1x=π 4 sin^{-1}x+cos^{-1} x=\pi  then x=
    Solution

  • Question 7
    1 / -0
    tan1 (c1xyc1y+x )+tan1 (c2c11+c2c1)+tan1 (c3c21+c3c2 )+ tan^{-1} \left (\dfrac{c_1\,x - y}{c_1\,y + x} \right ) + tan^{-1} \left ( \dfrac{c_2 - c_1}{1 + c_2c_1} \right ) + tan^{-1} \left (\dfrac{c_3 - c_2}{1 + c_3c_2}  \right ) + ..... +tan1(1cn )= + tan^{-1}\left (\dfrac{1}{c_n}  \right ) =
    Solution

  • Question 8
    1 / -0
    The value of tan1(xcosθ1xsinθ)cot1(cosθxsinθ)\tan^{-1}\Bigg(\dfrac{x\cos\theta}{1-x\sin\theta}\Bigg)-\cot^{-1}\Bigg(\dfrac{\cos\theta}{x-\sin\theta}\Bigg)  is
    Solution

  • Question 9
    1 / -0
    Let  $$\begin{vmatrix}tan^{-1}x & tan^{-1}2x & tan^{-1}3x\\ tan^{-1}3x & tan^{-1}x & tan^{-1}2x\\
    tan^{-1}2x & tan^{-1}3x & tan^{-1}x \end{vmatrix}$$= 0, then the number of values of x satisfying the equation is 
    Solution

  • Question 10
    1 / -0
    If cot1(15)=x\cot^{-1}\left(\dfrac {-1}{5}\right)=x then sinx=\sin x= ?
    Solution

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