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question_answer1) \[{{\cot }^{-1}}\left[ \frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}} \right]=\] [MNR 1986]
question_answer2) The principal value of \[{{\sin }^{-1}}\left[ \sin \left( \frac{2\pi }{3} \right) \right]\]is [IIT 1986]
question_answer3) If \[\theta ={{\tan }^{-1}}a,\varphi ={{\tan }^{-1}}b\] and \[ab=-1,\] then \[\theta -\varphi =\]
question_answer4) If \[\tan ({{\cos }^{-1}}x)=\sin \left( {{\cot }^{-1}}\frac{1}{2} \right)\], then x =
question_answer5) If \[{{\sin }^{-1}}x=\frac{\pi }{5}\] for some \[x\in (-1,\,1)\], then the value of \[{{\cos }^{-1}}x\] is [IIT 1992]
question_answer6) If\[{{\cos }^{-1}}p+{{\cos }^{-1}}q+{{\cos }^{-1}}r=\pi \]then\[{{p}^{2}}+{{q}^{2}}+{{r}^{2}}+2pqr=\] [Karnataka CET 2004]
question_answer7) \[\tan \left[ \frac{\pi }{4}+\frac{1}{2}{{\cos }^{-1}}\frac{a}{b} \right]+\tan \left[ \frac{\pi }{4}-\frac{1}{2}{{\cos }^{-1}}\frac{a}{b} \right]=\] [MP PET 1999]
question_answer8) \[{{\tan }^{-1}}\,\left[ \frac{\sqrt{1+{{x}^{2}}}+\sqrt{1-{{x}^{2}}}}{\sqrt{1+{{x}^{2}}}-\sqrt{1-{{x}^{2}}}} \right]=\]
question_answer9) The equation \[{{\sin }^{-1}}x-{{\cos }^{-1}}x={{\cos }^{-1}}\left( \frac{\sqrt{3}}{2} \right)\]has
question_answer10) \[\sum\limits_{m=1}^{n}{{{\tan }^{-1}}}\left( \frac{2m}{{{m}^{4}}+{{m}^{2}}+2} \right)\]is equal to
question_answer11) The number of real solutions of \[{{\tan }^{-1}}\sqrt{x(x+1)}+{{\sin }^{-1}}\sqrt{{{x}^{2}}+x+1}=\frac{\pi }{2}\]is [IIT 1999]
question_answer12) The equation \[2{{\cos }^{-1}}x+{{\sin }^{-1}}x=\frac{11\pi }{6}\]has [AMU 1999]
question_answer13) If \[{{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z=\pi \], then \[x+y+z\] is equal to [Kerala (Engg.) 2002]
question_answer14) \[2{{\tan }^{-1}}\left[ \sqrt{\frac{a-b}{a+b}}\tan \frac{\theta }{2} \right]=\] [Dhanbad Engg. 1976]
question_answer15) \[2{{\tan }^{-1}}(\cos x)={{\tan }^{-1}}(\text{cose}{{\text{c}}^{2}}x),\] then x = [UPSEAT 2002]
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