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Question
Show that tan 75° + cot 75° = 4
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Solution
tan 75° = tan(45° + 30°)
= `(tan45^circ + tan30^circ)/(1 - tan45^circ tan30^circ)`
= `(1 + 1/sqrt(3))/(1 - 1/sqrt(3))`
= `((sqrt(3) + 1)/sqrt(3))/((sqrt(3) - 1)/sqrt(3))`
= `(sqrt(3) + 1)/(sqrt(3) - 1)`
cot 75° = `1/tan75^circ`
= `(sqrt(3) - 1)/(sqrt(3) + 1)`
So, L.H.S = tan 75° + cot 75°
= `(sqrt(3) + 1)/(sqrt(3) - 1) + (sqrt(3) - 1)/(sqrt(3) + 1)`
= `((sqrt(3) + 1)^2 + (sqrt(3) - 1)^2)/((sqrt(3) - 1)(sqrt(3) + 1)`
= `(3 + 1 + 2sqrt(3) + 3 + 1 - 2sqrt(3))/(sqrt(3)^2 - 1^2)`
= `8/(3 - 1)`
= `8/2`
= 4
= R.H.S
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