Use the following diagram to help you find an exact value for tan 22.5^{0} without using a calculator.

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Suppose AC is of length 1

ADC = 45^{o} [Smaller angle of a right-angled isosceles triangle]

ADB = 135^{o} [Angles on a straight line]

ABC = 22.5^{o} [Angle sum of isoceles triangle ABD]

\( \tan ABC = \frac{ AC }{ BC} \)

AD = \( \sqrt(2) \) [Pythagoras]

BD = AD = \( \sqrt(2) \)

BC = DC + BD = \( 1 + \sqrt(2) \)

\( \therefore \tan ABC = \frac{1 }{ 1 + \sqrt(2) } \)

Multiply the top and bottom of this fraction by the conjugate surd \( 1 - \sqrt(2) \)

$$ \tan 22.5^o = \sqrt{2} - 1 $$

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