Problem 8 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In a tringle \(\Delta A B C\), prove that \((a-b)^{2} \cos ^{2}\left(\frac{C}{2}\right)+(a+b)^{2} \sin ^{2}\left(\frac{C}{2}\right)=c^{2}\)

Step-by-Step Solution

Answer

With the use of trigonometric identities including the Pythagorean Identity and Law of Cosines as well as some mathematical manipulation, it has been proven that the original expression is equal to \(c^{2}\).

Step 1: Expand the expression

Expand the square terms in the identity we are asked to prove. So \((a-b)^{2} \cos^{2}\left(\frac{C}{2}\right)+(a+b)^{2} \sin^{2}\left(\frac{C}{2}\right)\) becomes \(a^{2}\cos^{2}(C/2) + b^{2}\cos^{2}(C/2) - 2ab\cos^{2}(C/2) + a^{2}\sin^{2}(C/2) + b^{2}\sin^{2}(C/2) + 2ab\sin^{2}(C/2)\)

Step 2: Reorganize the expression

To simplify this expression, one should group the like terms. This gives us \((a^{2}+b^{2})\cos^{2}(C/2) + (a^{2}+b^{2})\sin^{2}(C/2) - 2ab\cos^{2}(C/2) + 2ab\sin^{2}(C/2)\)

Step 3: Apply the Pythagorean identity

Now, recall the Pythagorean Identity, \(\sin^{2}(\theta) + \cos^{2}(\theta) = 1\). One should then replace \(\cos^{2}(C/2) + \sin^{2}(C/2)\) by 1 in our expression. So the expression becomes \((a^{2}+b^{2}) - 2ab\cos^{2}(C/2) + 2ab\sin^{2}(C/2)\)

Step 4: Apply the Law of Cosines

Apply the Law of Cosines which states that \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Considering that \(C/2\) is half of the angle C in our expression and applying the double-angle formula, we find that \(\cos C = 2\cos^{2}(C/2) - 1\). Rearranging that gives us \(\cos^{2}(C/2) = (1 + \cos C)/2\). Likewise, the formula for \(\sin^{2}(C/2)\) is \((1 - \cos C)/2\). Substituting these values into our expression creates a match between our expression and the formula of the Law of Cosines, thus proving the identity \((a^{2}+b^{2}) - 2ab\cos^{2}(C/2) + 2ab\sin^{2}(C/2) = c^{2}\)

Key Concepts

Law of Cosines

The Law of Cosines is a fundamental principle in trigonometry, primarily used to determine a third side of a triangle when two sides and their included angle are known. It's like the Pythagorean Theorem, but applies to any triangle, not just right-angled ones. The standard form of the Law of Cosines is: \[ c^2 = a^2 + b^2 - 2ab \cos C \]where:

• \( c \) is the side opposite angle \( C \).
• \( a \) and \( b \) are the other two sides.
• \( C \) is the angle opposite side \( c \).

This law helps in various scenarios to find unknown side lengths or angles, especially when the triangle is not right-angled. Understanding this can be crucial in tackling geometry problems that involve triangles. From navigation to physics, the law finds extensive applications where precise calculations are required.

Pythagorean identity

The Pythagorean identity is one of the most famous identities in trigonometry. It stems from the Pythagorean Theorem and states:\[\sin^2{\theta} + \cos^2{\theta} = 1\]This identity is true for all angles \( \theta \) and is used extensively in proving other trigonometric equations and transformations. In practical terms:

• It implies that in a unit circle (a circle with radius 1), any point on the circle satisfies this identity.
• It is the basis for other identities such as the secant and cosecant identities.
• It simplifies expressions in trigonometry and converts between functions.

Replacing or reorganizing trigonometric expressions using this identity often makes calculations more manageable. It can transform complex equations into simpler forms, proving especially helpful in both pure and applied mathematics.

Trigonometric transformations

Trigonometric transformations involve converting one trigonometric function into another or simplifying trigonometric expressions using known identities. They're key to solving trigonometric equations efficiently.Two important transformations used in our provided solution are:

• The double angle formulas which split or merge angles, such as for cosine: \[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
• The half-angle formulas allow you to break down expressions involving half-angles, beneficial for functions involving \( C/2 \).

These transformations facilitate deeper insights into the properties of trig functions and help solve complex angular problems more easily. With practice, they become incredibly powerful tools for both students and professionals dealing with trigonometric contexts.