Question

Find T Ratios of 75 in:
1. cot 75

Answer 1

Answer:

tan 75°  =  tan (45° + 30°)

 =  (tan 45° + tan 30°)/ (1 - tan 45° tan 30°)

 =  (1 + (1/√3)) /(1 - 1(1/√3))

 =  [(√3 + 1)/√3] / [(√3 - 1)/√3]

 =  (√3 + 1)/(√3 - 1)

Multiply by (√3 + 1) on both numerator and denominator.

 =  (√3 + 1)2/(√32 - 12)

 =  (3 + 1 + 2√3) / (3 - 1)

 =  (4 + 2√3) / 2

 =  2 + √3   ------(1)

tan 15°  =  cot (90° - 15°)

tan 15°  =  cot 75°

Instead of finding the value of cot 75, let us find the value of tan 15.

tan 15°  =  tan (45° - 30°)

Using compound angle formula, we get

 =  (√3 - 1)/(√3 + 1)

Multiply by (√3 - 1) on both numerator and denominator.

 =  (√3 - 1)2/(√32 - 12)

 =  (3 + 1 - 2√3) / (3 - 1)

 =  (4 - 2√3) / 2

 =  2 - √3   ------(2)

(1) + (2)

tan 75° + cot 75°  =    2 + √3  +  2 - √3  

tan 75° + cot 75°  =  4

Hence proved.

Step-by-step explanation:

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