Is tan^2(pi/6)= 1/3 ?

The answer to whether tan^2(pi/6)= 1/3 is True

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prove tan^2(pi/6)= 1/3

prove

tan^{2} (\frac{π}{6}) = \frac{1}{3}

True

Solution steps

tan^{2} (\frac{π}{6}) = \frac{1}{3}

Manipulating left side

tan^{2} (\frac{π}{6})

Simplify

tan^{2} (\frac{π}{6}) : \frac{1}{3}

tan^{2} (\frac{π}{6})

Simplify

tan (\frac{π}{6}) : \frac{3}{3}

tan (\frac{π}{6})

Use the following trivial identity:

tan (\frac{π}{6}) = \frac{3}{3}

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

= (\frac{3}{3})^{2}

Simplify

(\frac{3}{3})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{3 ^{2}}

(3)^{2} : 3

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= \frac{3}{3 ^{2}}

Cancel the common factor:

3

= \frac{1}{3}

= \frac{1}{3}

= \frac{1}{3}

We showed that the two sides could take the same form

\Rightarrow True

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