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Question

Prove that
(i) tan-11-x22x+cot-11-x22x=π2

(ii) sintan-11-x22x+cos-11-x22x=1

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Solution

(i)
tan-11-x22x+cot-11-x22x=π2LHS=tan-11-x22x+cot-11-x22x =tan-11-x22x+π2-tan-11-x22x tan-1x+cot-1x=π2 =π2=RHS

(ii)
sintan-11-x22x+cos-11-x21+x2=1LHS=sintan-11-x22x+cos-11-x21+x2 =sinsin-11-x22x1+1-x22x+cos-11-x21+x2 tan-1x=sin-1x1+x2 =sinsin-11-x21+x2+cos-11-x21+x2 =sinπ2 sin-1x+cos-1x=π2 =1=RHS

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