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Question

Evaluate:

(i) cotsin-134+sec-143
(ii) sintan-1x+tan-11x for x<0
(iii) sintan-1x+tan-11x for x>0
(iv) cottan-1a+cot-1a
(v) cossec-1x+cosec-1x, x1

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Solution

(i)
cotsin-134+sec-143=cotsin-134+cos-134 sec-1x=cos-11x =cotπ2 sin-1x+cos-1x=π2=0

(ii)
sintan-1x+tan-11x=sintan-1-x+tan-1-1x x<0=sin-tan-1x-tan-11x=sin-tan-1x+tan-11x=sin-tan-1x+cot-1x tan-11x=cot-1x=-sintan-1x+cot-1x =-sinπ2 tan-1x+cot-1x=π2=-1

(iii)
sintan-1x+tan-11x=sintan-1x+cot-1x tan-1x=cot-11x =sinπ2 tan-1x+cot-1x=π2=1

(iv)
cottan-1a+cot-1a =cotπ2 tan-1x+cot-1x=π2=0

(v)
cossec-1x+cosec-1x =cosπ2 sec-1x+cosec-1x=π2=0

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