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NARRATOR: Exercise 7.

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Prove that 1 divided by
sec theta plus tan theta

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is equal to cos theta divided by
1 plus sin theta.

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We begin with the left-hand side,

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1 divided by sec theta
plus tan theta,

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and write this out as a division

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so that we can express sec theta
and tan theta as fractions.

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So we have 1 divided by 1 over cos
theta plus sin theta over cos theta.

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So that is 1 divided by 1 plus
sin theta all divided by cos theta,

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which is equivalent to

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1 multiplied by cos theta
divided by 1 plus sin theta,

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which equals cos theta

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divided by 1 plus sin theta,

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which is equal to

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the right-hand side,

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and so we have proven that

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1 divided by sec theta plus tan theta

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equals cos theta

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divided by 1 plus sin theta

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as was required.