Problem 42 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sin 105^{\circ}+\cos 105^{\circ}=\cos 45^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 105^{\circ}+\cos 105^{\circ}=\cos 45^{\circ} \) is verified.

Step 1: Simplify sin(105) using sine of sum identity

Sin(105) is equal to sin(60 + 45). We can simplify this expression using the formula for the sine of a sum, \( \sin(A+B) = \sin A \cos B + \cos A \sin B \). Therefore, \( \sin 105^{\circ} = \sin 60^{\circ} \cos 45^{\circ} + \cos 60^{\circ} \sin 45^{\circ} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 2: Simplify cos(105) using cosine of sum identity

Cos(105) is equal to cos(60 + 45). We can simplify this expression using the formula for the cosine of a sum, \( \cos(A+B) = \cos A \cos B - \sin A \sin B \). Therefore, \( \cos 105^{\circ} = \cos 60^{\circ} \cos 45^{\circ} - \sin 60^{\circ} \sin 45^{\circ} = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2} - \sqrt{6}}{4} \).

Step 3: Add the results

Add the results from step 1 and step 2 to verify the equality: \( \frac{\sqrt{6} + \sqrt{2}}{4} + \frac{\sqrt{2} - \sqrt{6}}{4} = \frac{\sqrt{2}}{2} \). Thus, \( \sin 105^{\circ}+\cos 105^{\circ}=\cos 45^{\circ} \).

Key Concepts

Sine of a Sum Identity

Understanding the sine of a sum identity can help simplify trigonometric expressions. This identity is represented as \( \sin(A+B) = \sin A \cos B + \cos A \sin B \). It illustrates how the sine of the sum of two angles, \( A \) and \( B \), can be expressed as the sum of products of sines and cosines of these angles.
For the angle \( 105^{\circ} \), the expression can break down to \( \sin(60^{\circ} + 45^{\circ}) \). By applying the formula:

• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)

By substituting these values, \( \sin 105^{\circ} \) simplifies to \( \frac{\sqrt{6} + \sqrt{2}}{4} \). This method helps you handle complex angles by using simpler components.

Cosine of a Sum Identity

The cosine of a sum identity is a fundamental part of trigonometry, aiding in the simplification of cosine expressions for angles that are sums of two known angles. Its formula is \( \cos(A+B) = \cos A \cos B - \sin A \sin B \). This tells us that the cosine of the sum can be rewritten as a difference of product of cosines and sines of the individual angles.
In the case of \( 105^{\circ} \), we rewrite it as \( \cos(60^{\circ} + 45^{\circ}) \). Applying this identity involves:

• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)

By substituting these values, we find \( \cos 105^{\circ} = \frac{\sqrt{2} - \sqrt{6}}{4} \). This shows how this identity is particularly useful for transformations involving the sum of angles.

Angle Addition Formulas

Angle addition formulas are the backbone of many trigonometric calculations, creating pathways for simplifying expressions involving sums of angles. They provide a strategic way to transform complex angle expressions into manageable components. These formulas include \( \sin(A+B) \) and \( \cos(A+B) \), allowing for a breakdown in terms of individual angle sines and cosines.
With the angle \( 105^{\circ} \), it's dissected into known angles \( 60^{\circ} \) and \( 45^{\circ} \). By utilizing the angle addition formulas:

• Break \( \sin 105^{\circ} \) with the sum identity: \( \sin(60^{\circ} + 45^{\circ}) \)
• Break \( \cos 105^{\circ} \) with the sum identity: \( \cos(60^{\circ} + 45^{\circ}) \)

These processes enable the conversion of complex trigonometric forms into simpler, more understandable terms. Such breakdowns facilitate error checking and deeper insight into trigonometric relationships, turning difficult angles into sums of familiar components.