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Proving Trigonometric Identities

So, you've memorized your fundamental identities, but how can you prove the more obscure ones? See how to apply the basic building blocks of trig to understand deeper relationships.

Proving Trigonometric Identities Warmup

\( \frac{\tan(\theta)}{\sin(\theta)} \) is equivalent to which of these?

If this equation is true...

\[ \sin^a(\theta) \cdot \csc^5(\theta) = \sin(\theta) \]

...then what is \(a\)?

Simplify:

\[ \sin^2(\theta) \csc^3(\theta) \cos^4(\theta) \sec^5(\theta) \tan^6(\theta) \cot^{7}(\theta) \]

How many ways are there to fill the blanks so the equation is true, choosing from the six basic trigonometric functions (sin, cos, tan, csc, sec, cot)?

\[ \frac{\sin^2(\theta) + 1}{\sin(\theta)} \]

Which expression is equivalent to the expression above?

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