Question

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Answer 1

Answer on Question #83570 – Math – Trigonometry

Question

Jana is proving that the following trigonometric identity is true:

$$\cos(-\theta) \tan \theta = \sin \theta$$

Which would be a correct first line of her proof?

$$\cos(\theta) \tan \theta = \sin \theta$$ $$\cos(-\theta) \tan \theta = \sin(-\theta)$$ $$\cos(-\theta) \tan(-\theta) = \sin(-\theta)$$ $$\cos(\theta) \tan(-\theta) = \sin(\theta)$$

Solution

Due to the symmetry of $\cos(\theta)$ (the cosine function is even, which means $\cos(-\theta) = \cos(\theta)$), the equality

$$\cos(-\theta) \tan(\theta) = \sin(\theta)$$

is transformed to

$$\cos(\theta) \tan(\theta) = \sin(\theta),$$

which is also valid due to the definition of the tangent function

$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.$$

Answer: the correct first line would be $\cos(\theta) \tan(\theta) = \sin(\theta)$.

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