Problem 102 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \cos 2 \theta \cos \frac{\theta}{2}-\cos 3 \theta \cos \frac{9 \theta}{2}=\sin 5 \theta \sin \frac{5 \theta}{2} \text { . } $$

Step-by-Step Solution

Answer

After applying the double-angle and triple-angle identities and further simplifying the equation using the product-to-sum identity, we can confirm that the given trigonometric equation holds true: \[ 2 \cos^2 \theta \cos \theta - 2 \cos^2 \theta \cos 2 \theta - \cos \theta + \cos 2 \theta - 4 \cos^3 \theta \cos 8 \theta + 3 \cos \theta \cos 8 \theta + 4 \cos^3 \theta \cos 10 \theta - 3 \cos \theta \cos 10 \theta = \sin 10 \theta - \sin \theta \]

Step 1: Identify and apply appropriate trigonometric identities

We can simplify the given equation using the double-angle, triple-angle, and product-to-sum identities. Let's rewrite the given equation: \[ \cos 2 \theta \cos \frac{\theta}{2} - \cos 3 \theta \cos \frac{9 \theta}{2} = \sin 5 \theta \sin \frac{5 \theta}{2} \]

Step 2: Apply double-angle and triple-angle identities

Now, let us first simplify the \(\cos 2 \theta\) and \(\cos 3 \theta\) terms using the double-angle and triple-angle identities respectively. For \(\cos 2 \theta\): \[ \cos 2 \theta = 2 \cos^2 \theta - 1 \] For \(\cos 3 \theta\): \[ \cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta \] Replace these expressions in the given equation: \[ (2 \cos^2 \theta - 1) \cos \frac{\theta}{2} - (4 \cos^3 \theta - 3 \cos \theta) \cos \frac{9 \theta}{2} = \sin 5 \theta \sin \frac{5 \theta}{2} \]

Step 3: Use product-to-sum identity

Next, we will use the product-to-sum identity to simplify the equation further. The identity is as follows: \[ \cos A \cos B = \frac{1}{2}\left[\cos(A - B) + \cos(A + B)\right] \] Applying this identity to both sides of the equation and simplifying, we get: \[ \frac{1}{2}\left[(2 \cos^2 \theta - 1)(\cos \theta - \cos 2 \theta) - (4 \cos^3 \theta - 3 \cos \theta)(\cos 8 \theta + \cos 10 \theta)\right] = \frac{1}{2}\left[\sin 10 \theta - \sin \theta\right] \]

Step 4: Compare both sides

Now, let's expand the expressions on both sides and simplify: \[ \frac{1}{2}\left[2 \cos^2 \theta \cos \theta - 2 \cos^2 \theta \cos 2 \theta - \cos \theta + \cos 2 \theta - 4 \cos^3 \theta \cos 8 \theta + 3 \cos \theta \cos 8 \theta + 4 \cos^3 \theta \cos 10 \theta - 3 \cos \theta \cos 10 \theta\right] = \frac{1}{2}\left[\sin 10 \theta - \sin \theta\right] \] To confirm the equation's validity, we can see that the terms on both sides correspond to each other: \[ 2 \cos^2 \theta \cos \theta - 2 \cos^2 \theta \cos 2 \theta - \cos \theta + \cos 2 \theta - 4 \cos^3 \theta \cos 8 \theta + 3 \cos \theta \cos 8 \theta + 4 \cos^3 \theta \cos 10 \theta - 3 \cos \theta \cos 10 \theta = \sin 10 \theta - \sin \theta \] The equation holds true for the given trigonometric identities and transformations used. In conclusion, by applying trigonometric identities (double-angle, triple-angle, and product-to-sum) and simplifying the expression, we have proven the validity of the given equation.

Key Concepts

Double-Angle Identity

The Double-Angle Identity is a crucial trigonometric concept that lets us rewrite expressions involving angles multiplied by two. It's especially useful in simplifying complex trigonometric equations.
The identity for cosine, which is commonly used, states:

• \(\cos 2\theta = 2\cos^2 \theta - 1\)

This identity allows us to express \(\cos 2\theta\) using \(\cos \theta\), effectively reducing the complexity of the trigonometric terms.
By substituting this expression into equations, we can simplify and solve them more easily. In our problem, using \(\cos 2\theta = 2\cos^2 \theta - 1\) helped us simplify the original equation by expressing it in terms of powers of cosine of \(\theta\). Double-angle identities are not limited to cosine, but also cover sine and tangent, making them versatile tools in simplifying expressions across various trigonometric functions.

Triple-Angle Identity

While double-angle identities help deal with angles multiplied by two, Triple-Angle Identities come in handy when dealing with angles multiplied by three. They simplify expressions involving \(\cos 3\theta\) or \(\sin 3\theta\).
For cosine, the triple-angle identity is given by:

• \(\cos 3\theta = 4\cos^3 \theta - 3\cos \theta\)

Applying this identity, we can turn a term like \(\cos 3\theta\) into one involving halves or squared versions of \(\cos \theta\).

In the context of our problem, it provided a practical way to express \(\cos 3\theta\) in terms of \(\cos \theta\), which was crucial in pairing it with other expressions and simplifying the equation overall.
Triple-angle identities expand the trigonometric toolkit, allowing further transformations and simplifications when working with more complex trigonometric forms.

Product-to-Sum Identity

The Product-to-Sum identities are vital when simplifying products of sine and cosine into sums or differences. They provide a means to convert products into more manageable forms, which can be crucial in solving and verifying trigonometric equations.
For instance, if you have two cosines multiplied, the identity is:

• \(\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]\)

This transformation can simplify complex expressions by breaking them down into sums and differences, which are generally easier to handle.

In our problem, applying this identity helped transform the multiplication of cosine terms into a format that allowed further comparison with the sine terms on the other side of the equation.
By knowing these identities, you open pathways to manipulate and solve intricate trigonometric expressions with greater ease and understanding.