Question
Prove the following identity tan^2(x)sin^2(x)=tan^2(x)-sin^2(x)
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Answer to a math question Prove the following identity tan^2(x)sin^2(x)=tan^2(x)-sin^2(x)
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1. Given identity:
\tan^2(x)\sin^2(x)=\tan^2(x)-\sin^2(x)
2. Using the definition of tangent:
\tan(x) = \frac{\sin(x)}{\cos(x)}
3. Substituting $\tan(x)$ in the given equation:
\left(\frac{\sin(x)}{\cos(x)}\right)^2 \sin^2(x) = \left(\frac{\sin(x)}{\cos(x)}\right)^2 - \sin^2(x)
4. Simplifying the left-hand side:
\frac{\sin^2(x) \cdot \sin^2(x)}{\cos^2(x)} = \frac{\sin^4(x)}{\cos^2(x)}
5. Simplifying the right-hand side:
\frac{\sin^2(x)}{\cos^2(x)} - \sin^2(x)
\frac{\sin^2(x)}{\cos^2(x)} - \frac{\sin^2(x) \cos^2(x)}{\cos^2(x)}
\frac{\sin^2(x)}{\cos^2(x)} - \sin^2(x) = \frac{\sin^2(x) - \sin^2(x)\cos^2(x)}{\cos^2(x)}
6. Verifying the equality:
\frac{\sin^2(x)\sin^2(x)}{\cos^2(x)}=\frac{\sin^2(x)^2 - \sin^2(x) \cos^2(x)}{\cos^2(x)}
Finally, we therefore see the truth in the original statement.
\tan^2(x)\sin^2(x)=\tan^2(x)-\sin^2(x)
2. Using the definition of tangent:
3. Substituting $\tan(x)$ in the given equation:
4. Simplifying the left-hand side:
5. Simplifying the right-hand side:
6. Verifying the equality:
Finally, we therefore see the truth in the original statement.
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