:D: 4) If tan-11+x2 - 1-x2 1+x2 +1-x2 = , Then prove that x2 - Sin 2 - Maths - Inverse Trigonometric Functions

Question

:D:

4) If tan - 1 1 + x 2   -   1 - x 2   1 + x 2  
+ 1 - x 2   =   ∞ ,
Then prove that x 2   -   S i n   2  
∞

Neha Sethi · Accepted Answer

Dear student
tan-11+x2-1-x21+x2+1-x2=α⇒1+x2-1-x21+x2+1-x2=tanα⇒1+x2-1-x2=tanα1+x2+1-x2⇒1+x2-1-x2=1+x2tanα+1-x2tanα⇒1+x21-tanα=1-x2tanα+1⇒-1+x21-x2=tanα+1tanα-1⇒1-x21+x2=1-tanα1+tanα⇒1-x21+x2=1-sinαcosα1+sinαcosα⇒1-x21+x2=cosα-sinαcosα+sinα⇒1-x21+x2=cosα-sinαcosα+sinα2⇒1-x21+x2=cos2α+sin2α-2sinα cosαcos2α+sin2α+2sinα cosα     ∵a±b2=a2+b2±2ab⇒1-x21+x2=1-sin2α1+sin2α    ∵sin2x+cos2x=1 and 2sinx cosx=sin2xOn comparing, we getx2=sin2α
Regards

:D: 4) If tan - 1 1 + x 2 - 1 - x 2 1 + x 2 + 1 - x 2 = ∞ , Then prove that x 2 - S i n 2 ∞ .

:D:

4) If $\tan^{- 1} \{\frac{\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}}{\sqrt{1 + x^{2}} + \sqrt{1 - x^{2}}}\} = \infty,$
Then prove that $x^{2} - S i n 2 \infty$ .