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Solution 4.3:8a

From Förberedande kurs i matematik 1

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m (Lösning 4.3:8a moved to Solution 4.3:8a: Robot: moved page)
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<center> [[Image:4_3_8a.gif]] </center>
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We rewrite
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<math>\text{tan }v</math>
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on the left-hand side as
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<math>\frac{\sin v}{\cos v}</math>, so that
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<math>\tan ^{2}v=\frac{\sin ^{2}v}{\cos ^{2}v}</math>
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If we then use the Pythagorean identity
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<math>\cos ^{2}v+\sin ^{2}v=1</math>
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and rewrite
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<math>\text{cos}^{\text{2}}v</math>
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in the denominator as
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<math>\text{1}-\text{sin}^{\text{2}}v\text{ }</math>, we get what we are looking for on the right-hand side. The whole calculation is
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<math>\tan ^{2}v=\frac{\sin ^{2}v}{\cos ^{2}v}=\frac{\sin ^{2}v}{1-\sin ^{2}v}</math>

Revision as of 10:37, 30 September 2008


We rewrite tan v on the left-hand side as sinvcosv, so that


tan2v=sin2vcos2v


If we then use the Pythagorean identity


cos2v+sin2v=1


and rewrite cos2v in the denominator as 1−sin2v , we get what we are looking for on the right-hand side. The whole calculation is


tan2v=sin2vcos2v=sin2v1−sin2v

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