Question

Prove that cot^–17 + cot^–18 + cot^–118 = cot^–13

Answer 1

We have cot^–17 + cot^–18 + cot^–118

= `tan^-1  1/7 + tan^-1  1/8 + tan^-1  1/18`  ......(since `cos^-1x = tan^-1  1/x`, if x > 0)

= `tan^-1 ((1/7 + 1/8)/(1 - 1/7 xx 1/8)) + tan^-1  1/18`  ......(since x . y = `1/7  1/8 < 1`)

= `tan^-1  3/11 + tan^-1  1/18`

= `tan^-1((3/11 + 1/18)/(1 - 3/11 xx 1/18))`  .....(since xy < 1)

= `tan^-1  65/195`

= `tan^-1  1/3`

= cot^–13