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Question
Prove that cot–17 + cot–18 + cot–118 = cot–13
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Solution
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
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