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Section 5.2 Proving Trig Identities
To verify an identity:
1. Simplify one side until it is identical to the other
side. Start with the more complicated side.
2. Simplify each side separately until both sides are
identical.
We write a formal proof of an identity by working
only with one side and show that it is identically
equal to the other side.
Strategies:
1. Change expressions to ones involving sines and
cosines using the basic trig identities.
2. Multiply an expression by a fraction equal to 1.
Since you are attempting to prove that the two
sides of the identity are equal, you may not use
any property that assumes the equation is valid
to begin with. In particular, you may not
multiply both sides by the same expression, nor
is it valid to assume a proportion and find the
cross product.
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More Strategies:
1. Combine fractions by combining them over a
common denominator.
2. Use the algebraic identity
to set up applications of the Pythagorean identities.
3. Always be mindful of the "target" expression, and
favor manipulations that bring you closer to your goal.
4. Disprove an identity by graphing and finding an xvalue for which the equation is not true.
Prove or disprove:
Look at the graphs of each expression first.
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How could this proof be shortened?
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Prove the identity:
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Use a graph to demonstrate that
Then verify the identity algebraically.
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Prove the identity:
Discuss problems 58, 59, & 61 on page 461.
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Prove the identity:
Now try to prove it again by working with
the other side.
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