Question

Simplify $$\sin A\cos A\tan A+\frac{2\sin A{\cos }^{3}A}{\sin 2A}$$

Short answer

Question: Simplify the trigonometric expression $\sin A\cos A(\cot A)+\frac{2\sin A{\cos }^{3}A}{2\sin A\cos A}$. Answer: $\cos A$

Step-by-step solution

Rewrite the expression in terms of sine and cosine

We can begin by rewriting the expression in terms of sine and cosine functions: $$\sin A\cos A(\frac{\sin A}{\cos A})+\frac{2\sin A{\cos }^{3}A}{2\sin A\cos A}$$

Simplify the expression

Now we can simplify the expression further by cancelling out common terms and performing simple calculations: $$\sin A\cos A\cdot \sin A+\frac{2\sin A{\cos }^{3}A}{2\sin A\cos A}$$

Cancel common terms in the fraction

In the second term, notice that we have common factors of 2, $\sin A$, and $\cos A$ in both the numerator and denominator, so we can cancel them: $$\sin A\cos A\cdot \sin A+\frac{\cancel{2}\cancel{\sin A}{\cos }^{\cancel{2}}A}{\cancel{2}\cancel{\sin A}\cancel{\cos A}}$$ This leaves us with: $$\sin A\cos A\cdot \sin A+{\cos }^{2}A$$

Apply the Pythagorean identity for sine and cosine

Recall that the Pythagorean identity for sine and cosine is given by: $${\sin }^{2}A+{\cos }^{2}A=1$$ We can use this identity to simplify the expression further. Since the first term is $\sin A\cos A\cdot \sin A={\sin }^{2}A\cos A$, we can rewrite the expression as: $${\sin }^{2}A\cos A+{\cos }^{2}A$$ Now, we can factor out $\cos A$ from both terms: $$\cos A({\sin }^{2}A+{\cos }^{2}A)$$ Using the Pythagorean identity, we can replace the expression inside the parentheses with 1: $$\cos A(1)=\cos A$$ So the original expression simplifies to: $$\cos A$$

Key concepts

Sine and Cosine Simplification

Understanding how to simplify expressions using the sine and cosine functions can be immensely beneficial. The fundamental trigonometric functions, sine ($\sin$) and cosine ($\cos$), are ratios of the sides of a right triangle to its hypotenuse.

In many trigonometric expressions, you'll encounter combinations of sine and cosine. For instance, multiplying and dividing by these functions.

Here are some straightforward steps to perform simplification:

• Rewrite any trigonometric ratio like $\tan A=\frac{\sin A}{\cos A}$ in terms of sine and cosine.
• Search for common terms that can be simplified or cancelled in a fraction.
• Organize and combine like terms, which can make further simplification possible.

By applying these methods, you can see a clearer path towards simplifying complex trigonometric expressions.

Pythagorean Identity

The Pythagorean identity is a bedrock principle of trigonometry. It's expressed as: $${\sin }^{2}A+{\cos }^{2}A=1$$This identity is derived from the Pythagorean theorem and holds for any angle $A$.

Why is this useful?

• It provides a direct relationship between sine and cosine, which is useful for simplifying expressions with these terms.
• When combined, these terms sum to 1, providing a shortcut in simplification problems.
• This identity often appears implicitly in problems and can drastically reduce the complexity of an expression.

For example, within a problem, ${\sin }^{2}A+{\cos }^{2}A=1$ can replace complex segments to make calculations simpler. Recognizing when to employ the Pythagorean identity is key to streamlining trigonometric expressions.

Trigonometric Expression Simplification

Trigonometric expression simplification involves breaking down complex expressions into more manageable forms. This simplification often involves strategic use of trigonometric identities.

Key approaches include:

• Utilize basic identities, like the Pythagorean identity, which was already discussed.
• Cancel out common terms in the expression. Look for factors that are present in both the numerator and denominator.
• Factor out common elements that can combine terms into simpler expression forms.
• Always watch out for substitutions using identities. For example, substituting ${\sin }^{2}A+{\cos }^{2}A$ with $1$ when applicable can simplify the expression significantly.

Adapting these strategies helps streamline otherwise cumbersome algebraic expressions by reducing them to a format that is much easier to compute and understand.