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Find the value of cot squared π given csc squared π is equal to 25 over nine.
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Now what weβre absolutely not going to do is solve the equation csc squared π equals 25 over nine for π.
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Instead, weβre going to recall some trigonometric identities.
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The first identity we recall is that sin squared π plus cos squared π is equal to one.
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And weβre going to derive the next identity, though it is absolutely fine to simply quote it.
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Weβre going to divide each term here by sin squared π.
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sin squared π divided by sin squared π is equal to one.
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Then, we recall that cot π is equal to one over tan π.
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And tan π is sin π over cos π.
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So its reciprocal cot π must be equal to cos π over sin π.
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This means that cos squared π over sin squared π must be equal to cot squared π.
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And then, since we know that csc π is equal to one over sin π, we can say that this is equal to csc squared π.
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Now, itβs absolutely fine to quote this identity one plus cot squared π equals csc squared π.
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To find an identity for cot squared π, Iβm going to subtract one from both sides.
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And we find that cot squared π is always equal to csc squared π minus one.
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Now, in our question, weβre told that csc squared π is 25 over nine.
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So we can say that cot squared π here must be equal to 25 over nine minus one.
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To evaluate this, letβs write one as nine over nine.
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And when we subtract nine over nine from 25 over nine, we get 16 over nine.
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And we find that when csc squared π is equal to 25 over nine, cot squared π is 16 over nine.