Problem 289 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \cos (A+B+C)=\cos A \cos B \cos C, \text { show that } 8 \sin (B+C) \sin (C+A) \sin (A+B)=-\sin 2 A \sin 2 B \sin 2 C \text { . } $$

Step-by-Step Solution

Answer

By applying the sin and cos rules to the original equation, you can simplify and rearrange it to get the desired equation, i.e. \(8 \sin(B+C) \sin(C+A) \sin(A+B)=-\sin 2A \sin 2B \sin 2C\). This proves the given statement.

Step 1: Apply Sine Rule

Apply the sin(phi+psi) rule to the left-hand side of the equation. The rule is given as \( \sin (\phi + \psi) = \sin \phi \cos \psi + \cos \phi \sin \psi \). You can split up \(\sin(B+C)\), \(\sin(C+A)\), and \(\sin(A+B)\) into their respective components.

Step 2: Apply Cosine Rule

Apply the \(\cos(\phi+\psi)\) rule to the \(\cos(A + B + C)\) term in the given equation. The rule is given by \(\cos(\phi + \psi) = \cos \phi \cos \psi - \sin \phi \sin \psi \). This will allow you to express \(\cos(A + B + C)\) in terms of \(\cos A \cos B \cos C\).

Step 3: Simplify the Equation

Simplify the equation by clever factorization and simplification using common trigonometric identities such as \(\sin 2x = 2 \sin x \cos x \). You should be able to simplify the equation down in such a way that the LHS of the equation resembles the RHS of the target equation.

Step 4: Proof of Equality

Rearrange and factorize the LHS in such a way that you get the RHS of the original equation i.e. \(-\sin 2A \sin 2B \sin 2C\). This completes the proof

Key Concepts

Sine Rule

The Sine Rule is a fundamental concept in trigonometry, particularly useful for solving problems involving non-right triangles. It allows us to find unknown sides or angles in a triangle when certain elements are known. The Sine Rule states:

• For any triangle with angles A, B, and C and respective sides a, b, and c, the following holds true:

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This equation can be reversed to help find unknown angles if the sides are known:

• \( \sin A = \frac{a \cdot \sin B}{b} \) or \( \sin A = \frac{a \cdot \sin C}{c} \)

In the context of the problem, the Sine Rule was applied by breaking down complex sine expressions. Using trigonometric identities such as \(\sin(\phi + \psi) = \sin \phi \cdot \cos \psi + \cos \phi \cdot \sin \psi\), you can express the sine of sum angles as products or sums of simpler functions. This is particularly useful in transforming the original equation into a more factorable form.

Cosine Rule

The Cosine Rule is another essential tool in trigonometry. It relates the lengths of the sides of a triangle to the cosine of one of its angles. This rule is particularly useful when dealing with non-right triangles, just like the Sine Rule. The formula for the Cosine Rule is:

• For a triangle with sides a, b, and c, opposite to angles A, B, and C respectively:

\[c^2 = a^2 + b^2 - 2ab \cos C\]So, if you know two sides and an included angle, you can find the third side. Alternatively, if you know all sides, you can find the angles. The connection to the current problem comes from using the identity:

• \( \cos(\phi + \psi) = \cos \phi \cos \psi - \sin \phi \sin \psi \)

This identity assists in expressing more complicated cosine expressions into manageable equations, thereby simplifying the proof process of given trigonometric equations. Essentially, by utilizing these rules, you can translate complex trigonometric functions into familiar arithmetic operations.

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions of an unknown angle. Solving these requires an understanding of trigonometric identities and rules such as the Sine and Cosine Rules. Trigonometric equations can often be transformed using known identities:

• \( \sin 2x = 2 \sin x \cos x \)
• \( \cos 2x = \cos^2 x - \sin^2 x \)

In the exercise, the objective was to show that two complex trigonometric expressions are equal. This typically involves:

• Applying identities to simplify expressions.
• Clever factorization for easier manipulation.
• Rearranging terms to reach the desired equality.

The goal is usually to reduce one side to match the other side in terms of known trigonometric expressions—achieving this by systematic simplification and strategic use of identities creates the proof requested in trigonometric problems. Such problems hone skills in recognizing relationships between angles and their sine or cosine values, often leading to insightful patterns and solutions.