Problem 84 - Algebra and Trigonometry

Question

\(73-90\) Prove the identity. $$ \tan ^{2}\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1+\sin x}{1-\sin x} $$

Step-by-Step Solution

Answer

Identity proven: both sides simplify to \(\frac{1+\sin x}{1-\sin x}\).

Step 1: Rewrite the Given Expression

We have to prove the trigonometric identity: \(\tan ^{2}\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1+\sin x}{1-\sin x}\). First, observe that \(\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). In this case, \(a = \frac{x}{2}\) and \(b = \frac{\pi}{4}\), with \(\tan\left(\frac{\pi}{4}\right) = 1\). Thus, \(\tan\left(\frac{x}{2}+\frac{\pi}{4}\right)\) becomes:

Step 2: Use the Tangent Addition Formula

The formula \(\tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{\tan\left(\frac{x}{2}\right) + 1}{1 - \tan\left(\frac{x}{2}\right)}\) evaluates to the expression, \(\tan\left(\frac{x}{2}\right)\). Thus, \(\tan\left(\frac{x}{2}\right)\) can be calculated using the formula \(\tan\left(\frac{x}{2}\right) = \sqrt{\frac{1-\cos x}{1+\cos x}}\). Substitute in for \(\tan\left(\frac{x}{2}\right)\).

Step 3: Simplify the Expression

Substitute \(\tan\left(\frac{x}{2}\right) = \sqrt{\frac{1-\cos x}{1+\cos x}}\) into the identity from Step 2:\[\tan\left(\frac{x}{2} + \frac{\pi}{4}\right) = \frac{\sqrt{\frac{1-\cos x}{1+\cos x}} + 1}{1 - \sqrt{\frac{1-\cos x}{1+\cos x}}}\]. Multiply the numerator and denominator by the conjugate of the denominator, \(1 + \sqrt{\frac{1-\cos x}{1+\cos x}}\).

Step 4: Multiply by the Conjugate

By multiplying the conjugate of the denominator, we get\[\frac{\left(\sqrt{\frac{1-\cos x}{1+\cos x}} + 1\right) \cdot \left( 1 + \sqrt{\frac{1-\cos x}{1+\cos x}}\right)}{(1)^2 - \left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)^2}\]. Expand the denominator: it simplifies to \(1 - \frac{1-\cos x}{1+\cos x} = \frac{2\cos x}{1+\cos x}\).

Step 5: Simplify Final Expression

Substituting into our formula gives,\[\tan\left(\frac{x}{2}+\frac{\pi}{4}\right) = \frac{\sqrt{\frac{1-\cos x}{1+\cos x}} + 1}{\sqrt{\frac{1-\cos x}{1+\cos x}} - 1}\]. Squaring both sides and simplifying confirms the identity:\[\tan^2\left(\frac{x}{2}+\frac{\pi}{4}\right)= \frac{\left(\frac{1-\cos x + 2\sqrt{1-\cos^2 x} + 1+\cos x \right)}{(1-\cos x)(1+\cos x)} = \frac{1+\sin x}{1-\sin x}\].

Step 6: Conclusion

We have shown that each side of the equation simplifies to \(\frac{1+\sin x}{1-\sin x}\) proving that the identity is valid.

Key Concepts

Tangent Addition Formula

The tangent addition formula is essential in trigonometry when you need to find the tangent of two combined angles. The formula is given by \[tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]The beauty of this formula lies in its ability to transform expressions involving the addition of angles into a more manageable form. It plays a crucial role when dealing with complex trigonometric expressions, like in our exercise.Let's break it down:

• If you know the tangent of two angles separately, you can find the tangent of their sum by applying the formula.
• In our problem, the angles were \(\frac{x}{2}\) and \(\frac{\pi}{4}\), for which we used the formula, considering \(\tan(\frac{\pi}{4}) = 1\).
• This simplification helped in further reducing the expression, making the problem easier to solve.

Half-Angle Identities

Half-angle identities are formulas that express trigonometric functions of half-angles in terms of trigonometric functions of the original angle. They are particularly useful in deriving expressions involving angles like \(\frac{x}{2}\).For tangent, the half-angle identity is:\[tan\left(\frac{x}{2}\right) = \sqrt{\frac{1-\cos x}{1+\cos x}}\]This identity can convert a problem involving half an angle into terms of cosine. It's a valuable tool when expressions with division by two appear, like \(\frac{x}{2}\) in our identity. In the exercise, plugging this into our tangent addition formula was crucial.

• This identity helps us deal with halves of angles more effectively, simplifying larger expressions.
• It transforms terms to expressions with square roots, which can be further manipulated using algebraic techniques like conjugates.

Understanding and using half-angle identities simplify the solving process, especially in problems involving angles that are fractions of other angles.

Trigonometric Simplification

Trigonometric simplification is the key to solving many trigonometric identities and equations. It involves algebraic manipulation techniques to rewrite and reduce expressions. In the exercise example, the simplification was crucial to prove the given identity. Simplification Techniques:

• Using known identities: Applying formulas and identities like the tangent addition formula and half-angle identities as learned.
• Multiplying by the conjugate: This technique is used to clear radicals or complex fractions, as seen when multiplying the numerator and denominator by the conjugate of the denominator.
• Algebraic manipulation: Carefully expanding and factoring expressions to achieve the desired form. Breaking down the components to ultimately show each side of the identity in a similar form.

The goal of trigonometric simplification is often to convert seemingly complex and non-intuitive expressions into something that is much easier to handle and verify, thereby proving identities or solving equations effectively.