Question

question 1 (1 point)
the six addition and subtraction compound angle formulas may be used to create all equivalent double angle formulas for sine, cosine and tangent.
true
false

Explanation:

The double - angle formulas for sine (\(\sin2\alpha = 2\sin\alpha\cos\alpha\)), cosine (\(\cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha = 2\cos^{2}\alpha - 1=1 - 2\sin^{2}\alpha\)) and tangent (\(\tan2\alpha=\frac{2\tan\alpha}{1 - \tan^{2}\alpha}\)) can be derived from the addition formulas. For example, using the formula \(\sin(A + B)=\sin A\cos B+\cos A\sin B\), when \(A = B=\alpha\), we get \(\sin2\alpha=\sin(\alpha+\alpha)=\sin\alpha\cos\alpha+\cos\alpha\sin\alpha = 2\sin\alpha\cos\alpha\). Similarly, for cosine \(\cos(A + B)=\cos A\cos B-\sin A\sin B\), when \(A = B=\alpha\), \(\cos2\alpha=\cos(\alpha+\alpha)=\cos\alpha\cos\alpha-\sin\alpha\sin\alpha=\cos^{2}\alpha-\sin^{2}\alpha\). And for tangent \(\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\), when \(A = B=\alpha\), \(\tan2\alpha=\frac{\tan\alpha+\tan\alpha}{1-\tan\alpha\tan\alpha}=\frac{2\tan\alpha}{1 - \tan^{2}\alpha}\).

Answer:

True