Question

Prove each identity. $$\frac{\sin 2 \alpha+1}{\cos \alpha+\sin \alpha}=\sin \alpha+\cos \alpha$$

Short answer

\(\sin \alpha + \cos \alpha\) is proven to be equal to \(\frac{\sin 2\alpha + 1}{\cos \alpha + \sin \alpha}\) through the use of double angle and sum to product identities.

Step-by-step solution

Use the Double Angle Identity for Sine

Start by replacing the term \(\sin 2\alpha\) with the double angle identity, which is \(\sin 2\alpha = 2\sin \alpha \cos \alpha\). The left side of the equation becomes \(\frac{2\sin \alpha \cos \alpha + 1}{\cos \alpha + \sin \alpha}\).

Separate into Two Fractions

Break down the expression into two separate fractions: \(\frac{2\sin \alpha \cos \alpha}{\cos \alpha + \sin \alpha} + \frac{1}{\cos \alpha + \sin \alpha}\).

Simplify the First Fraction

In the first fraction \(\frac{2\sin \alpha \cos \alpha}{\cos \alpha + \sin \alpha}\), the \(\cos \alpha\) term cancels with the denominator in one of the terms, leaving us with \(2\sin \alpha\).

Rewrite the Second Fraction

In the second fraction \(\frac{1}{\cos \alpha + \sin \alpha}\), recognize that this is the same as multiplying by the reciprocal of \(\cos \alpha + \sin \alpha\).

Use the Sum to Product Identity

Apply the sum to product identity to recognize that \(\cos \alpha + \sin \alpha\) can be rewritten using \(2\sin\left(\frac{\alpha}{2} + \frac{\pi}{4}\right)\).

Apply the Reciprocal Identity

Using the reciprocal identity, we recognize that \(2\sin\left(\frac{\alpha}{2} + \frac{\pi}{4}\right)\) is the reciprocal of the second term in our original fractions which allows for simplification.

Combine the Results

We now combine the results from step 3 and the simplified fraction from step 6 to get \(2\sin \alpha + \frac{1}{2\sin\left(\frac{\alpha}{2} + \frac{\pi}{4}\right)} = \sin \alpha + \cos \alpha\), proving the initial identity.

Key concepts

Double Angle Identities

Trigonometry includes a special set of identities known as double angle identities. These are expressions that simplify functions involving angles that are twice as large as a single angle in question. For example, the double angle identity for sine is \[ \sin 2\alpha = 2\sin \alpha \cos \alpha \.\] This means that instead of directly calculating \( \sin \) of twice an angle, we can express it using the sine and cosine of the original angle, \( \alpha \).

In the context of proving trigonometric identities, these double angle identities are powerful tools. They allow us to transform complex expressions into simpler forms that are easier to manipulate. As seen in the exercise solution, replacing \( \sin 2\alpha \) with its double angle equivalent was the first step in simplifying the given equation.

Sum to Product Identities

Another set of identities in trigonometry is the sum to product identities, which are used to convert the sum or difference of two trigonometric functions into a product of sines and cosines. For instance, the sum of sine and cosine can be expressed as a product by using the identity \[ \cos \alpha + \sin \alpha = 2\sin\left(\frac{\alpha}{2} + \frac{\pi}{4}\right) \.\]

These identities are particularly helpful when simplifying complex trigonometric expressions, especially when we encounter sums or differences in denominators or numerators. In our exercise, the sum to product identity enables us to rewrite \( \cos \alpha + \sin \alpha \) in a form that is more conducive to finding a reciprocal, which is essential in the simplification process described in steps 4 through 6.

Trigonometry Proof

Executing a trigonometry proof often involves a sequence of logical steps where identities are used to demonstrate the equality of two expressions. A proof in trigonometry is not just a way of solving a problem; it's a method to show the underlying relationships between trigonometric functions. Proofs help in understanding the how and why of trigonometric principles.

In the given exercise, the proof starts with given trigonometric expressions and systematically applies identities such as double angle and sum to product. It's crucial to recognize which identity to use and when to use it, as seen from step 1 to step 7. Through each step, we at once simplify the expression and maintain its equivalence to the original, until we prove the desired identity.

Simplifying Trigonometric Expressions

The process of simplifying trigonometric expressions is essential for solving trigonometry problems effectively. It involves using identities to break down complex expressions into more manageable parts. This includes canceling out terms, finding common denominators, converting sums to products, or finding reciprocals.

As highlighted in the step-by-step solution, simplification may start with segregation of terms (as in step 2), followed by cancellation (step 3), and then utilizing more complex identities to manipulate the expression further (steps 5 and 6). The goal is always to reduce the complexity of the expression while preserving its original value, ultimately proving the equality in step 7.

Tip for Simplification

It's often helpful to have a clear strategy before starting. Begin by identifying terms that can be combined or canceled, and anticipate how an identity can bring you closer to the final, simplified form of the expression.