Answers posted by thanvigandhi_1

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Prove that $\cos\bigg\{2\cot^{-1}\sqrt{\large\frac{1-x}{1+x}}\bigg\}+x=0$

answered Feb 20, 2013

Toolbox: put $ x=cos2\theta \Rightarrow 2\theta=cos^{-1}x$ \( 1-cos2\theta=2\: sin...

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Prove that $4\tan^{-1}\large\frac{1}{5}-\tan^{-1}\frac{1}{70}+\tan^{-1}\frac{1}{99}=\large\frac{\pi}{4}$

answered Feb 20, 2013

Toolbox: $ 2tan^{-1}x=tan^{-1}\large\frac{2x}{1-x^2}$, |x|<1 \( tan^{-1}x-tan^{...

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If $ \cos^{-1}\frac{x}{a}+\cos^{-1}\frac{y}{b}=\alpha,$prove that $\frac{x^2}{a^2}-\frac{2xy}{ab}\cos \alpha+\frac{y^2}{b^2}=\sin^2\alpha.$

answered Feb 20, 2013

Toolbox: $ cos^{-1}x+cos^{-1}y=cos^{-1} \bigg[ xy+\sqrt{1-x^2}. \sqrt{1-y^2} \bigg] $ ...

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If $ \sin^{-1}x+\sin^{-1}y+\sin^{-1}z=\pi,$ prove that $x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=2xyz$

answered Feb 20, 2013

Toolbox: $cosx-cosy=2sin\large\frac{x+y}{2}.cos\large\frac{y-x}{2}$ \(1-sin^2x=cos...

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Prove that $\sin2[\cot^{-1}\{\cos(\tan^{-1}x)\}]=2\frac{\sqrt{x^2+1}}{x^2+2}$

answered Feb 19, 2013

Toolbox: $ cosy=\large\frac{1}{\sqrt{1+tan^2y}}$ $ sin2A=2sinAcosA$ \(ta...

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In any Triangle ABC, if $ A=\tan^{-1}2\;and\;B=\tan^{-1}3,$prove that $C=\frac{\pi}{4}$

answered Feb 19, 2013

Toolbox: In $ \Delta\: ABC \: A+B+C = \pi $ \( tan(A+B)=\large\frac{tanA+tanB}{1-t...

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If $ x= cosec \lfloor\tan^{-1}\{\cos(\cot^{-1}(\sec(\sin^{-1}a)))\}\rfloor\; $ and $ y=\sec\lfloor\cot^{-1}\{\sin(\tan^{-1}(cosec(\cos^{-1}a)))\}\rfloor\;$ where $\;a\;\in [0,1]$ Find the relationship between $x$ and $y$ in terms of $a$

answered Feb 19, 2013

Toolbox: $ sin^{-1}a=sec^{-1}\large\frac{1}{\sqrt{1-a^2}}$ \( cos^{-1}a=cosec^{-1}...

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Prove that :$\cot^{-1}\bigg[\frac{ab+1}{a-b}\bigg]+\cot^{-1}\bigg[\frac{bc+1}{b-c}\bigg]+\cot^{-1}\bigg[\frac{ca+1}{c-a}\bigg]=0$

answered Feb 19, 2013

Toolbox:$ cot^{-1}x=tan^{-1}\frac{1}{x}$$ tan^{-1}tanx=x$\(tan(A-B)=\frac{tanA-tanB}{1+tanAtanB}...

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Solve for x $ \sin^{-1}\frac{2\alpha}{1+\alpha^2}+ \sin^{-1}\frac{2\beta}{1+\beta^2}=2\tan^{-1}x.$

answered Feb 19, 2013

Toolbox: $ sin2A= \large\frac{2tanA}{1=tan^2A}$ \( tan^{-1}x+tan^{-1}y=tan^{-1}\la...

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Prove: $\cos^{-1}x-\cos^{-1}y=\cos^{-1}\bigg[xy+\sqrt{1-x^2}.\sqrt{1-y^2}\bigg]$

answered Feb 19, 2013

Toolbox:$ cos(A-B) = cosA\: cosB+ sinA\: sinB$$1-cos^2A=sin^2A$Let \( cos^{-1}x=A\: and \: cos^{...

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If $ \tan^{-1}a+\tan^{-1}b+\tan^{-1}c=\pi,\;prove\;that\;a+b+c=abc.$

answered Feb 19, 2013

Toolbox: $ tan^{-1}a+tan^{-1}b=tan^{-1}\large\frac{a+b}{1-ab}$ \( tan (\pi\: -\the...

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Prove that $ \tan^{-1}\large\frac{yz}{xr}+ \tan^{-1}\frac{zx}{yr}+ \tan^{-1}\frac{xy}{zr}=\frac{\pi}{2}\;where\;x^2+y^2+z^2=r^2$

answered Feb 19, 2013

Toolbox: $ tan^{-1}x+tan^{-1}y=tan^{-1} \bigg( \large\frac{x+y}{1-xy} \bigg) $ L.H.S. Using...

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Solve for x:$\tan^{-1}(x+1)+\tan^{-1}x+\tan^{-1}(x-1)=\tan^{-1}3.$

answered Feb 19, 2013

Toolbox: $ tan^{-1}x + tan^{-1}y = tan^{-1}\large\frac{x + y}{1 - xy}, xy < 1 $ ...

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Solve: $\sin\lfloor2\cos^{-1}\{\cot(2\tan^{-1}x)\}\rfloor=0$

answered Feb 19, 2013

Toolbox:put $\tan^{-1}x=\theta$$\Rightarrow x = \tan\theta$$\tan2\theta = \frac{2\tan\theta}{1-\tan^...

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If $ \cos^{-1}\Large\frac{x}{2}+\normalsize\cos^{-1}\large\frac{y}{3}=\theta ,\;\normalsize prove\;that\;9x^2-12xy\cos\theta +4y^2=36\sin^2\theta $

answered Feb 19, 2013

Toolbox: $sin(cos^{-1}x)=\sqrt{1-x^2}=cos(sin^{-1}x)$ \( cos (A+B)=cosa\: cosB-sin...

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Find the value of $\; \tan^{-1}\bigg(\tan{\frac{5\pi}{6}}\bigg)+\cos^{-1}\bigg(\cos{\frac{13\pi}{6}}\bigg)$

answered Feb 19, 2013

Toolbox:Principal interval of $\tan$ is $(-\large\frac{\pi}{2},\large\frac{\pi}{2})$Principal interv...

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Find the value of $\tan^{-1}\bigg(\tan{\frac{2\pi}{3}}\bigg)$

answered Feb 19, 2013

Toolbox:Principal interval of tan is $(-\large\frac{\pi}{2},\large\frac{\pi}{2})$$\tan(\pi-\theta)=-...

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Find the real solution of the equation
$\tan^{-1}\sqrt{x(x+1)}+\sin^{-1}\sqrt{x^2+x+1}=\frac{\pi}{2}$

answered Feb 19, 2013

Toolbox:Domain of $\sin^{-1}x=[-1,1]$Domaim of $\sqrt{x}$ is $x$ should be $\geq\:0$$\tan^{-1}\sqrt{...

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Find the value of the expression $\sin \bigg(2\tan^{-1}{\frac{1}{3}}\bigg)+\cos \big(\tan^{-1}2\sqrt 2\big)$

answered Feb 19, 2013

Toolbox:$ 2\tan^{-1}x=\tan^{-1}\large\frac{2x}{1-x^2} $$\tan^{-1}x=\sin^{-1}\large\frac{x}{\sqrt{1+x...

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If $2\tan^{-1}(\cos x)=\tan^{-1}(2cosecx)$,then what is the value of $x$?

answered Feb 19, 2013

Toolbox:$ 2\tan^{-1}y=\tan^{-1}\large\frac{2y}{1-y^2}\:\:|y|<1$$1-\cos^2x=\sin^2x$By taking $y=\c...

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Show that $\cos\bigg(2\tan^{-1}{\frac{1}{7}}\bigg)=\sin\bigg(4\tan^{-1}{\frac{1}{3}}\bigg)$

answered Feb 19, 2013

Toolbox: $ 2tan^{-1}x=tan^{-1}\large\frac{2x}{1-x^2} $ \( tan^{-1}x=cos^{-1}\large...

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Solve the following equation $\cos\big(\tan^{-1}x\big)=\sin\bigg(\cot^{-1}{\frac{3}{4}}\bigg)$

answered Feb 19, 2013

Toolbox:$\tan^{-1}x=\cos^{-1}\large\frac{1}{\sqrt{1+x^2}}$$\cot^{-1}x=\sin^{-1}\large\frac{1}{\sqrt{...

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Prove that $ \sin^{-1}{\frac{8}{17}}+\sin^{-1}{\frac{3}{5}}=\sin^{-1}{\frac{77}{85}}$

answered Feb 19, 2013

Toolbox: $ sin^{-1}x+sin^{-1}y=sin^{-1} \bigg( x\sqrt{1-y^2}+y\sqrt{1-x^2} \bigg)$ ...

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Show that $\sin^{-1}{\frac{5}{13}}+\cos^{-1}{\frac{3}{5}}=\tan^{-1}{\frac{63}{16}}$

answered Feb 19, 2013

Toolbox: $ sin^{-1}x=tan^{-1}\bigg( \frac{x}{\sqrt{1-x^2}} \bigg) $ \( cos^{-1}x=t...

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What does $\tan^{-1}{\frac{1}{4}}+\tan^{-1}{\frac{2}{9}}$ reduce to?

answered Feb 19, 2013

Toolbox:$\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}\:\:\:xy<1$$\tan^{-1}x=\sin^{-1}\frac{x}{...

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Find the value of $4\tan^{-1}{\large \frac{1}{5}}$$-\tan^{-1}{\large\frac{1}{239}}$

answered Feb 19, 2013

Toolbox:$ 2tan^{-1}x=tan^{-1}\frac{2x}{1-x^2}$ \( tan^{-1}x-tan^{-1}y=tan^{-1}\frac{x-y}{1+xy}\...

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If $a_1,a_2,a_3,......a_n$ is an arithmetic progression with common difference d,then evaluate the following expression $\tan \bigg( \tan^{-1}\bigg(\frac{d}{1+a_1a_2}\bigg)+\tan^{-1}\bigg(\frac{d}{1+a_2a_3}\bigg)+\tan^{-1}\bigg(\frac{d}{1+a_3a_4}\bigg)+........+\tan^{-1}\bigg(\frac{d}{1+a_{n-1}a_n}\bigg) \bigg) $

answered Feb 18, 2013

Toolbox: $ d=a_2-a_1=a_3-a_2=a_4-a_3=......a_n-a_{n-1}$ \( tan^{-1}\frac{x-y}{1+xy...

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Which of the following is the principal value branch of $\cos^{-1}x$?

answered Feb 18, 2013

Ans (C) $ [0, \pi ] $

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Which of the following is the principal value branch of $\text{cosec}^{-1}x$?

answered Feb 18, 2013

Ans (D) $\bigg[ -\large\frac{\pi}{2} \:\large \frac{\pi}{2} \bigg] - \{0\} $

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If $3\tan^{-1}x+\cot^{-1}x=\pi$ then $x$ equals

answered Feb 18, 2013

Toolbox:$ \tan^{-1}x+\cot^{-1}x=\large\frac{\pi}{2} $$\tan^{-1}1=\large\frac{\pi}{4}$ Ans (B) 1 ...

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The value of $\sin^{-1}\bigg(\cos\frac{33\pi}{5}\bigg)$ is

answered Feb 18, 2013

Toolbox:$cos(2n\pi+\theta)=cos\theta$Principal interval of cos is $[0,\pi]$$cosx=sin(\frac{\pi}{2}-x...

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The domain of the function $\cos^{-1}(2x-1)$ is

answered Feb 18, 2013

Toolbox:Domain of $\cos^{-1}x$ is $[-1,1]$Ans (A) [0, 1]By taking $2x-1$ in the place of $x$ we get$...

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The domain of the function defined by $f(x)=\sin^{-1}\sqrt{x-1}$ is

answered Feb 18, 2013

Toolbox:Domain of $\sin^{-1}x$ is $[-1,1]$ Ans (A) [1,2] By taking $\sqrt{x-1}$, in the pl...

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If $\cos\bigg(\sin^{-1}\frac{2}{5}+\cos^{-1}x\bigg)=0$,then x is equal to

answered Feb 18, 2013

Toolbox: $\cos \large\frac{\pi}{2}=0$ or $\cos^{-1}0=\large\frac{\pi}{2}$$ \sin^{-1}x+\cos^{-1}x=\la...

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The value of $\sin(\tan^{-1}(.75))$ is equal to

answered Feb 18, 2013

Toolbox: $2tan^{-1}x=tan^{-1}\large\frac{2x}{1-x^2},\:\:|x|<1$ \(tan^{-1}x=sin^...

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The value of $\cos^{-1}\bigg(\cos\frac{3\pi}{2}\bigg)$ is equal to:

answered Feb 18, 2013

Toolbox:Principal interval of cos is $[0,\pi]$$\sin 0=0$$\cos^{-1}0=\large\frac{\pi}{2}$$\cos\large\...

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The value of the expression $2\sec^{-1}2+\sin^{-1}\frac{1}{2}$ is:

answered Feb 18, 2013

Toolbox:$ \sin^{-1}\large\frac{1}{2}=\large\frac{\pi}{6},$$\sec^{-1}2=\large\frac{\pi}{3}$Ans (B) $\...

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If $\tan^{-1}x+\tan^{-1}y=\frac{4\pi}{5},then\;\cot^{-1}x+\cot^{-1}y$ equals:

answered Feb 18, 2013

Toolbox:$\tan^{-1}x+\cot^{-1}x=\large\frac{\pi}{2}$Ans - (A) $ \frac{\pi}{5} $ From the above form...

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If $\sin^{-1}\frac{2a}{1+a^2} +\cos^{-1}\frac{1-a^2}{1+a^2}=\tan^{-1}\frac{2x}{1-x^2},where\; a,x,\in\;[0,1],$then the value of x is

answered Feb 18, 2013

Toolbox: Put $ a=tanA \Rightarrow A = tan^{-1}a$ \( \large\frac{2tanA}{1+tan^2A}=s...

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The value of $\cot\bigg(\cos^{-1}\frac{7}{25}\bigg)$ is

answered Feb 18, 2013

Toolbox:$\cos\theta=x\:\Rightarrow\:\cot\theta=\large\frac{x}{\sqrt{1-x^2}}$ Ans - (D)Put $\cos^{-...

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The value of the expression $\tan\bigg(\frac{1}{2}\cos^{-1}{\frac{2}{\sqrt 5}}\bigg)$ is:

answered Feb 18, 2013

Toolbox:$\tan\large\frac{\theta}{2}=\sqrt{\large\frac{1-\cos\theta}{1+\cos\theta}}$Take $ cos^{-1}\l...

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If $|\;x\;|\leq 1,then\;2\tan^{-1}x+\sin^{-1}\frac{2x}{1+x^2}$ is equal to:

answered Feb 18, 2013

Toolbox:$ \large\frac{2\tan\theta}{1+\tan^2\theta}=\sin2\theta$Ans - (A) $4\tan^{-1}x$ put $...

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If $\cos^{-1}\alpha+\cos^{-1}\beta+\cos^{-1}\gamma=3\pi$,then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ equals\[(A)\;0\quad(B)\quad 1\quad(C)\quad 6\quad(D)\quad 12\]

answered Feb 18, 2013

Toolbox: Principal interval of cos is [$0,\pi$] $cos\pi=-1$ Ans - ( C ) ...

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The number of real solutions of the equation$\sqrt{1+\cos2x}=\sqrt 2\cos^{-1}(\cos x)\;in\begin{bmatrix}\frac{\Large \pi}{2},\pi\end{bmatrix}$

answered Feb 18, 2013

Toolbox:$1+\cos2x=2\cos^2x$ Ans - A, no real solution. $ \sqrt{1+\cos2x} = \sqrt {2 \cos^2...

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If $\cos^{-1}x\;>\;\sin^{-1}x,$ then \begin{array}{1 1}(A)\quad\frac{1}{\sqrt 2}0\end{array}

answered Feb 18, 2013

Toolbox: cosx>sinx if $x\:\in[0,\large\frac{\pi}{4}]$ Ans -(B) \(0,\le...

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The principal value of $\cos^{-1}\bigg(-\Large {\frac{1}{2}}\normalsize \bigg)$ is_____________.

answered Feb 18, 2013

Toolbox: The range of the principal value of $\cos^{-1}x$ is $\left [ 0,\pi \right ]$ $\cos (\...

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If $\cos(\tan^{-1}x+\cot^{-1}\sqrt 3)=0,$then the value of $x$ is _______________.

answered Feb 18, 2013

Toolbox:$\tan^{-1}x+\cot^{-1}x=\large\frac{\pi}{2} $$\cos \large\frac{\pi}{2} = 0\:\:\:or\:cos^{-1}0...

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The value of $\sin^{-1}\bigg(\sin\frac{3\pi}{5}\bigg)$ is ________________.

answered Feb 18, 2013

Toolbox:Principal interval of sin is $ [\large\frac{\pi}{2},-\large\frac{\pi}{2}]$$\sin(\pi-\theta)=...

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The set of values of $\sec^{-1}\big(\frac{1}{2}\big)$ is ______________.

answered Feb 18, 2013

Toolbox:Domain of $\sec^{-1}x\:is\:\:(-\infty,-1]\cup[1,\infty)$ Ans:Not defined $ sec^{-1...

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The principal value of $ \tan^{-1}\sqrt 3 $ is _______________.

answered Feb 18, 2013

Toolbox:The range of the principal value of $\; \tan^{-1}x$ is $\left (- \large\frac{\pi}{2},\large\...

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