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Question

Evaluate each of the following:

(i) cot-113-cosec-1-2+sec-123

(ii) cot-12cossin-132

(iii) cosec-1-23+2cot-1-1

(iv) tan-1-13+cot-113+tan-1sin-π2

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Solution

(i)
cot-113-cosec-1-2+sec-123=cot-1cotπ3-cosec-1cosec-π6+sec-1secπ6=π3+π6+π6=2π3

(ii)
cot-12cossin-132=cot-12cossin-1sinπ3=cot-12cosπ3=cot-12×12=cot-11=cot-1tanπ4=π4

(iii)
cosec-1-23+2cot-1-1=cosec-1cosec-π3+2cot-1cot3π4=-π3+2×3π4=-π3+3π2=7π6

(iv)
tan-1-13+cot-113+tan-1sin-π2=tan-1tan-π6+cot-1cotπ3+tan-1-1=tan-1tan-π6+cot-1cotπ3+tan-1tan-π4=-π6+π3-π4=-π12

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