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  • fill in the blanks If y equal to 2 tan invers x plus sin inverse 2x divided 1 plus x square for all x, then blank less than y less than blank

Fill in the blanks if y= 2 \tan^{-1}x+\sin^{-1}\frac{2x}{1+x^{2}} for all x,  then ______<y<_____.

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Fill in the blanks if y= 2 \tan^{-1}x+\sin^{-1}\frac{2x}{1+x^{2}} for all x,  then -2\pi< y< 2\pi

y= 2 \tan^{-1}x+\sin^{-1}\frac{2x}{1+x^{2}}

We know that,

2 \tan^{-1}p=\sin^{-1}\frac{2x}{1+x^{2}}

so

2 \tan^{1}x+\sin^{-1}\frac{2x}{1+x^{2}}=2 \tan^{1}x+2\tan^{-1}x

=4 tan-1 x

So, y = 4 tan-1 x

As, principal value of tan-1 x is  \left (-\frac{\pi}{2},\frac{\pi}{2} \right )

So,  4 \tan^{-1}x\epsilon \left ( -2\pi,2\pi \right )

Hence, -2π < y < 2π

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infoexpert24

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