Professor Turner wrote the expression below on the chalkboard for his team at the Research Center. He asked the scientists there
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For , on the left we have
, and on the right,
(where we use the double angle identity: )
Suppose the relation holds for :
Then for , the left side is
So we want to show that
On the left side, we can combine the fractions:
Recall that
so that we can write
(another double angle identity: )
Then recall that
which lets us consolidate the numerator to get what we wanted:
and the identity is established.