tan(75°) = tan (30 + 45)
Now,
[tex] \tan( A + B) = \frac{\tan A + \tan B}{1 - \tan A × \tan B} \\ [/tex]
[tex] \tan( 30 + 45) = \frac{\tan 30 + \tan 45}{1 - \tan 30 × \tan 45}\\ [/tex]
Here, we know that
tan 30 = √3 ÷ 3 (Rational)
tan 45 = 1
[tex] tan(75) = \frac{ \frac{ \sqrt{3} }{3} + 1 }{1 - \frac{ \sqrt[]{3} }{3} } \\ [/tex]
[tex] = \frac{ \sqrt{3} + 3}{3 - \sqrt{3} } \times \frac{3 + \sqrt{3} }{3 + \sqrt{3} } \\ [/tex]
[tex] = \frac{3 \sqrt{3} + 3 + 9 + 3 \sqrt{3} }{6} \\ [/tex]
[tex] = \frac{12 + 6 \sqrt{3} }{6} \\ [/tex]
Cancel 6 we get,
2 + √3 or 2 + 1.732 = 3.732
Thus, The exact value of tan 75 is 2 + √3
-TheUnknownScientist 72
