Without using a calculator, show that
(i) tan 75° = 2 + √3,
(ii) sec2 75° = 4 tan 75°
Haha…wicked! I love these questions. For some, it would be a pain…why cannot use the calculator? Afterall, that’s what the calculator is for, isn’t it? Well, anyway, it is not the answer that is important, it is the thought process. This is deceptively simple. Let’s get at it…
Now, when we think about this, there is not much into it that we can do. 75° is not a special angle. So how? Well, 75° may not be a special angle but it is the sum of two special angles: 30° and 45°! So, now we can translate
tan 75° = tan (30° + 45°). So now, we will use the tan (A + B) rule.
Now, tan 30° = 1/√3 and tan 45° = 1
So,
Rationalising the denominator, we get,
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