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Question
Find the value of tan `(7pi)/12`
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Solution
tan `(7pi)/12 = 7 xx 180/12`
= 7 × 15°
= 105°
tan(105°) = tan(90° + 15°)
= – cot 15°
= `- 1/tan15^circ`
= `- 1/(tan(45^circ - 30^circ))`
= `- 1/((tan45^circ - tan30^circ)/(1 + tan45^circ* tan30^circ))`
= `- (1 + tan45^circ* tan30^circ)/(tan45^circ - tan30^circ)`
= `- (1 + (1) * (1/sqrt(3)))/(1 - 1/sqrt(3))`
= `- ((sqrt(3)+ 1)/sqrt(3))/((sqrt(3) - 1)/sqrt(3))`
= `- (sqrt(3) + 1)/(sqrt(3) - 1)`
= `- (sqrt(3) + 1)/(sqrt(3) - 1) xx - (sqrt(3) + 1)/(sqrt(3) + 1)`
= `- ((sqrt(3) + 1)^2)/((sqrt(3))^2 - 1^2)`
= `- ((3 + 2sqrt(3) + 1))/(3 - 1)`
tan(105°) = `- (4 + 2sqrt(3))/2`
= `- (2 + sqrt(3))`
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