Evaluate the following: cot - 1 1 3 - cosec - 1 ( - 2 ) + sec - 1 2 3

Given that, cot - 1 1 3 - cosec - 1 ( - 2 ) + sec - 1 2 3 + sec - 1 sec π 6 = cot - 1 cot π 3 + cosec - 1 cosec π 6 + sec - 1 sec π 6 = π 3 + π 6 + π 6 = 2 π 3

Evaluate the following: cot - 1 2 cos sin - 1 3 2

Given that, cot - 1 2 cos sin - 1 3 2 = cot - 1 2 cos sin - 1 sin π 3 We know that, sin - 1 sin x = x , x ∈ 0 , π . = cot - 1 2 cos π 3 = cot - 1 2 × 1 2 = cot - 1 ( 1 ) = cot - 1 cot π 4 = π 4

Evaluate the following: cosec - 1 - 2 3 + 2 cot - 1 ( - 1 ) .

Given that, cosec - 1 - 2 3 + 2 cot - 1 ( - 1 ) = cosec - 1 cosec - π 3 + 2 cot - 1 cot 3 π 4 = - π 3 + 2 × 3 π 4 = - π 3 + 3 π 2 = 7 π 6

Evaluate the following: tan - 1 - 1 3 + cot - 1 1 3 + tan - 1 sin - π 2

Given that, tan - 1 - 1 3 + cot - 1 1 3 + tan - 1 sin - π 2 = tan - 1 tan - π 6 + cot - 1 cot π 3 + tan - 1 ( - 1 ) = tan - 1 tan - π 6 + cot - 1 cot π 3 + tan - 1 tan - π 4 = - π 6 + π 3 - π 4 = - π 12

Evaluate each of the following: sin - 1 sin π 6 .

Given that, sin - 1 sin π 6 We know that, sin - 1 sin x = x , where x ∈ - π 2 , π 2 Since, π 6 ∈ - π 2 , π 2 ⇒ sin - 1 sin π 6 = π 6

We know sin - 1 sin θ = θ where - π 2 ≤ θ ≤ π 2 We have sin - 1 sin 7 π 6 = sin - 1 sin π + π 6 = sin - 1 sin - π 6 = - π 6

Given that, sin - 1 sin 5 π 6 We know that, sin sin - 1 θ = θ if - π 2 ≤ θ ≤ π 2 sin - 1 sin 5 π 6 = sin - 1 sin π - π 6 = sin - 1 sin π 6 = π 6

Given that, sin - 1 sin 13 π 7 We know that, sin sin - 1 θ = θ if - π 2 ≤ θ ≤ π 2 sin - 1 sin 13 π 7 = sin - 1 sin 2 π - π 7 = sin - 1 sin - π 7 = - π 7