The given equation are illustrations of trigonometry identities. The value of [tex]\sin^2\theta + \sin^4\theta[/tex] is 1
Given that:
[tex]\cos \theta + \cos^2 \theta = 1[/tex]
In trigonometry:
[tex]\sin^2 \theta + \cos^2\theta = 1[/tex]
Rewrite as:
[tex]\sin^2 \theta = 1 - \cos^2\theta[/tex]
Make [tex]\cos \theta[/tex] the subject in [tex]\cos \theta + \cos^2 \theta = 1[/tex]
[tex]\cos\theta = 1 - \cos^2 \theta[/tex]
Compare [tex]\sin^2 \theta = 1 - \cos^2\theta[/tex] and [tex]\cos\theta = 1 - \cos^2 \theta[/tex]
[tex]\sin^2 \theta = \cos \theta[/tex]
So:
[tex]\sin^2\theta + \sin^4\theta[/tex] becomes
[tex]\sin^2\theta + \sin^4\theta = \sin^2\theta + (\sin^2\theta)^2[/tex]
Substitute [tex]\sin^2 \theta = \cos \theta[/tex]
[tex]\sin^2\theta + \sin^4\theta = cos\theta + (cos\theta)^2[/tex]
[tex]\sin^2\theta + \sin^4\theta = cos\theta + cos^2\theta[/tex]
Recall that: [tex]\cos \theta + \cos^2 \theta = 1[/tex]
This means:
[tex]\sin^2\theta + \sin^4\theta = 1[/tex]
Hence, the value of [tex]\sin^2\theta + \sin^4\theta[/tex] is 1
Read more about trigonometry identities at:
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