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NARRATOR: Exercise 13.

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Show that t equals tan
of 67.5 degrees

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satisfies the quadratic equation
t squared minus 2t minus 1 equals 0

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and hence find its exact value.

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Using the double angle formula
for tan,

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that is tan of 2x is equal to

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2 times tan x divided by
1 minus tan squared x,

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we can see that tan of 135 degrees,
which is 2 times 67.5 degrees,

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is equal to 2 times
tan of 67.5 degrees

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divided by 1 minus
tan of 67.5 degrees all squared.

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Substituting the fact that
t equals tan of 67.5 degrees,

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we see that tan of 135 degrees

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is equal to 2t divided by
1 minus t squared.

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Now, tan of 135 degrees we can
evaluate using the unit circle.

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So 135 degrees is an angle
in the second quadrant

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that is 45 degrees away
from the x-axis.

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And so we can relate tan of
45 degrees with tan of 135 degrees

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using symmetry.

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And see that tan of 135 degrees is
equal to negative tan of 45 degrees.

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Using a "special triangle", we can
see that tan of 45 degrees equals 1,

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so therefore tan of 135 degrees
is equal to negative 1.

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So negative 1 equals 2t
divided by 1 minus t squared

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which means that negative 1
minus t squared is equal to 2t

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or t squared minus 1 is equal to 2t

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so t squared minus 2t

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minus 1 equals 0.

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And so we've established that
t squared minus 2t minus 1 equals 0.

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And so we can go on
to solve this equation for t

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in order to determine the exact value
of tan of 67.5 degrees.

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So we first attempt to factorise
this quadratic.

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Factors of negative 1
that add to negative 2.

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And since there are no such factors,
we go to the quadratic formula.

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So t is equal to 2 plus or minus

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the square root of 4 minus 4 times 1
times negative 1 all divided by 2,

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which is 2 plus or minus
the square root of 4 plus 4 over 2,

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which is 2 plus or minus
the square root of 8 over 2.

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The square root of 8
simplifies to 2 root 2.

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And dividing everything through by 2,

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we get 1 plus or minus

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the square root of 2.

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But t, which is tan of 67.5 degrees,

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must be positive,

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and we know this from the unit circle

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that tan of 67.5 degrees

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is going to be a positive value.

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So therefore tan of 67.5 degrees

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must be equal to

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1 plus the square root of 2.