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How do you prove #tantheta/cottheta=tan^2theta#?

Answer 1

Caroline Adams

see below

#tantheta/cottheta = tan^2theta#

Left Side: #=tantheta/cottheta #

#=tan theta*1/cot theta#

#=tan theta tan theta#

#=tan^2theta#

#:.=# Right Side

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Answer 2

Brooklyn Anderson

To prove (\frac{\tan(\theta)}{\cot(\theta)} = \tan^2(\theta)), we start with the left-hand side: [\frac{\tan(\theta)}{\cot(\theta)} = \frac{\frac{\sin(\theta)}{\cos(\theta)}}{\frac{\cos(\theta)}{\sin(\theta)}}]

Next, we simplify by multiplying the numerator by the reciprocal of the denominator: [\frac{\tan(\theta)}{\cot(\theta)} = \frac{\sin(\theta)}{\cos(\theta)} \times \frac{\sin(\theta)}{\cos(\theta)}]

Simplifying further, we get: [\frac{\tan(\theta)}{\cot(\theta)} = \frac{\sin^2(\theta)}{\cos^2(\theta)}]

Using the trigonometric identity (\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}), we see that the left-hand side is equal to (\tan^2(\theta)). Therefore, the identity is proven.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.