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Find the value of x if  \sec^2(tan^{-1}(-1))+\csc^{2}(cot^{-1}x)=12

Option: 1

3


Option: 2

-3


Option: 3

3, -3


Option: 4

None of these


Answers (1)

best_answer

We have learnt that

Function f (f-1 ( x )), where f(x) is a trigonometric function

\begin{array} {l}\mathrm{1.\;\;\sin(\sin^{-1}(x))=x} \quad\quad\quad \;\mathrm{for\;all\;x\in[-1,1] }\\\mathrm{2.\;\;\cos(\cos^{-1}(x))=x} \quad\quad\quad \mathrm{for\;all\;x\in[-1,1]}\\\mathrm{3.\;\;\tan(\tan^{-1}(x))=x} \;\;\;\quad\quad \mathrm{for\;all\;x\in\mathbb{R}}\\\mathrm{4.\;\;\cot(\cot^{-1}(x))=x} \quad\quad\quad \mathrm{for\;all\;x\in\mathbb{R}} \\\mathrm{5.\;\;\sec(\sec^{-1}(x))=x} \quad\quad\quad \mathrm{for\;all\;x\in\mathbb{R}-(-1,1)}\\\mathrm{6.\;\;\csc(\csc^{-1}(x))=x} \quad\quad\quad \mathrm{for\;all\;x\in\mathbb{R}-(-1,1)}\end{array}

Now,

\\\sec^2(tan^{-1}(-1))+\csc^{2}(cot^{-1}x)=12\\ \{ 1+ tan^{2}(tan^{-1}(-1))\}+\{ 1+cot^{2}(cot^{-1}x)\}=12\\ 1+(-1)^2+1+x^2=12\\ x=3,-3

Posted by

avinash.dongre

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