Find T Ratios of 75 in:
1. cot 75
Answers
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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