Answer:
H: 2
Step-by-step explanation:
[tex]\frac{\sqrt{1-\cos ^2\left(x\right)}}{\sin \left(x\right)}+\frac{\sqrt{1-\sin ^2\left(x\right)}}{\cos \left(x\right)}[/tex]
LCM is [tex]\cos \left(x\right)\sin \left(x\right)[/tex]
Adjust fractions:
[tex]\frac{\sqrt{1-\cos ^2\left(x\right)}\cos \left(x\right)}{\sin \left(x\right)\cos \left(x\right)}+\frac{\sqrt{1-\sin ^2\left(x\right)}\sin \left(x\right)}{\cos \left(x\right)\sin \left(x\right)}[/tex]
Combine:
[tex]\frac{\sqrt{1-\cos ^2\left(x\right)}\cos \left(x\right)+\sqrt{1-\sin ^2\left(x\right)}\sin \left(x\right)}{\sin \left(x\right)\cos \left(x\right)}[/tex]
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We will take a look at [tex]\sqrt{1-\cos ^2\left(x\right)}\cos \left(x\right)[/tex].
Using the identity that [tex]\cos ^2\left(x\right)+\sin ^2\left(x\right)=1[/tex], we know that [tex]1-\cos ^2\left(x\right)=\sin ^2\left(x\right)[/tex].
Therefore, [tex]\sqrt{1-\cos ^2\left(x\right)}\cos \left(x\right)=\sqrt{\sin ^2\left(x\right)}\cos \left(x\right)=\sin \left(x\right)\cos \left(x\right)[/tex]
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Similarly, [tex]1-\sin ^2\left(x\right)=\cos ^2\left(x\right)[/tex] so
[tex]\sqrt{1-\sin ^2\left(x\right)}\sin \left(x\right)=\sqrt{\cos ^2\left(x\right)}\sin \left(x\right)=\cos \left(x\right)\sin \left(x\right)[/tex]
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We will finally end up with
[tex]\frac{\sin \left(x\right)\cos \left(x\right)+\cos \left(x\right)\sin \left(x\right)}{\sin \left(x\right)\cos \left(x\right)}[/tex]
Add:
[tex]\frac{2\sin \left(x\right)\cos \left(x\right)}{\sin \left(x\right)\cos \left(x\right)}[/tex]
Cancel out sin(x):
[tex]\frac{2\cos \left(x\right)}{\cos \left(x\right)}[/tex]
Cancel out cos(x):
[tex]2[/tex]
