Question

Prove the following identities using only the definitions of the trigonometric functions, the Pythagorean identity, or the identities for sines and cosines of sums of angles. a. $\cos 2x=2{\cos }^{2}x-1$ b. $\sin 3x=A{\sin }^{3}x+B\sin x$, for what values of $A$ and $B$ ? c. $\sec \theta +\tan \theta =\tan (\frac{\theta }{2}+\frac{\pi }{4})$

Short answer

a. True by the cosine double angle formula. b. $A=-4$ and $B=3$ by the triple angle formula. c. True by the half-angle and tangent addition formulas.

Step-by-step solution

Part a: Applying Cosine Double Angle Formula

Use the identity $\cos 2x=2{\cos }^{2}x-1$, which is a direct application of the cosine double angle formula.

Part b: Applying the Triple Angle Formula

$\sin 3x=\sin (2x+x)$. Apply the sine addition formula $\sin (a+b)=\sin a\cos b+\cos a\sin b$. In this case, $a=2x$ and $b=x$, so $\sin 3x=\sin 2x\cos x+\cos 2x\sin x$. Apply the double angle formulas for sine and cosine, resulting in $3\sin x-4{\sin }^{3}x$. So, $A=-4$ and $B=3$.

Part c: Applying Half-Angle and Addition Formulas

Firstly, rewrite the right hand side using the tangent addition formula, $\tan (a+b)=(\tan a+\tan b)/(1-\tan a\tan b)$. Then, replace $\tan \theta /2$ with its equivalence in terms of secant from the half-angle formula. Simplify to obtain $sec\theta +tan\theta$.

Key concepts

Cosine Double Angle Formula

The Cosine Double Angle Formula is a powerful tool in trigonometry that helps simplify expressions involving angles that are twice as large as the initial angles. The formula is given by:

• $\cos 2x=2{\cos }^{2}x-1$
• It can also be expressed as $\cos 2x=1-2{\sin }^{2}x$
• Another version involves both sine and cosine: $\cos 2x={\cos }^{2}x-{\sin }^{2}x$

Understanding this formula can simplify complex problems such as finding the exact cosine value for an angle that is twice another, or analyzing oscillations and waves.
To derive the formula, start with the Pythagorean identity ${\sin }^{2}x+{\cos }^{2}x=1$. Using this, substitute ${\sin }^{2}x$ with $1-{\cos }^{2}x$ in one of the variations of the formula, and you can easily switch among these three forms.
Each form proves useful in different scenarios depending on which function you start with. This so-called double angle identity is a cornerstone in simplifying trigonometric expressions during problem-solving.

Sine Addition Formula

The Sine Addition Formula is an essential identity in trigonometry used to calculate the sine of an angle that is a sum of two other angles. The formula is expressed as:

• $\sin (a+b)=\sin a\cos b+\cos a\sin b$

This identity helps break down complex angles into simpler components, which is particularly useful when evaluating the sine of multiple angle sums.
Let's explore its application with an example: to find $\sin 3x$, you could split it into $\sin (2x+x)$. Applying the Sine Addition Formula results in $\sin 2x\cos x+\cos 2x\sin x$.
Further simplifying using known double angle formulas gives us $3\sin x-4{\sin }^{3}x$, identifying suitable coefficients as required in specific trigonometric identities. Thus, knowing the fundamentals of addition formulas broadens your understanding in various branches of physics and engineering, particularly when faced with waveforms and signals.

Tangent Half-Angle Formula

The Tangent Half-Angle Formula is another critical tool in trigonometry. It facilitates calculations involving half-angles, particularly useful in integrals and limits. The formula states:

• $\tan \left(\frac{\theta }{2}\right)=\frac{1-\cos \theta }{\sin \theta }=\frac{\sin \theta }{1+\cos \theta }$

The formula provides two equivalent forms, offering flexibility depending on the given problem context.
In practice, this helps when solving problems like determining exact tangent values or simplifying expressions involving tangent, secant, or cosine. In the specific exercise we analyzed, transforming the half-angle expression assists in determining the identity $\sec \theta +\tan \theta$. By combining half-angle identities with tangent addition formulas, you can simplify and solve complex trigonometric expressions effectively.
Mastering the Tangent Half-Angle Formula is crucial for higher-level mathematics involving calculus, as it forms the bedrock for solving numerous trigonometric integrals and series.