Problem 28 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{\cos A \operatorname{cosec} A-\sin A \sec A}{\cos A+\sin A}=\operatorname{cosec} A-\sec A $$

Step-by-Step Solution

Answer

The equation \( \frac{\cos A \operatorname{cosec} A-\sin A \sec A}{\cos A+\sin A}=\operatorname{cosec} A-\sec A \) holds true as both sides of the equation are equal after applying various trigonometric identities and simplifications.

Step 1: Replace Trigonometric Terms

Replace \( \csc A \) and \( \sec A \) with \( \frac{1}{\sin A} \) and \( \frac{1}{\cos A} \) respectively. We then have \( \frac{\cos A * \frac{1}{\sin A} - \sin A * \frac{1}{\cos A}}{\cos A + \sin A} = \frac{1}{\sin A} - \frac{1}{\cos A} \).

Step 2: Simplify the Given Expression

Simplify above to get \( \frac{\frac{\cos A}{\sin A} - \frac{\sin A}{\cos A}}{\cos A + \sin A} = \cot A - \tan A \). Further simplification gives \( =\frac{\frac{\cos^2 A - \sin^2 A}{\sin A \cos A}}{\cos A + \sin A} = \frac{\cos 2A}{\sin 2A} \).

Step 3: Convert into Trigonometric Identities

Convert \( \frac{\cos 2A}{\sin 2A} \) into cot using the fact that cotA = \( \frac{\cos A}{\sin A} \). Hence we have \( \cot 2A \).

Step 4: Use Double Angle Formulas

By using double angle formulas, we have \( \cot 2A \) equals to \( \frac{1 - \tan^2 A}{2 \tan A} \). Further simplification gives \( \frac{1 - (\sin^2 A / \cos^2 A)}{2 \sin A/ \cos A} \).

Step 5: Reverting Back to Initial Trigonometric Identities

We now replace \( \tan A \), \( \csc A \) and \( \sec A \) back with their original terms, giving us \( \frac{1 - \frac{1}{\csc^2 A}}{2 / \sec A} \), simplifying further gives \( \frac{\csc^2 A - 1}{2 \csc A} = \csc A - \sec A \) proving the initial equation.

Key Concepts

Trigonometric Formulas

Trigonometric formulas are fundamental tools in mathematics that allow us to simplify and solve various problems involving angles and their relationships. They encompass a wide array of equations such as the Pythagorean identities, reciprocal identities, and co-function identities. Students often encounter these formulas in the context of right-angled triangles, where they relate the angles to the ratios of the triangle's sides.

For example, the primary trigonometric functions sine (\r\(\sin\)), cosine (\r\(\cos\)), and tangent (\r\(\tan\)) have reciprocal identities given by cosecant (\r\(\csc\)), secant (\r\(\sec\)), and cotangent (\r\(\cot\)), respectively. Understanding these relationships is crucial, as seen in the provided exercise where the cosecant and secant are expressed as reciprocals of sine and cosine.

Additionally, mastering these formulas enables students to transition into more complex concepts such as the double angle formulas, which are pivotal in the simplification of trigonometric expressions.

Double Angle Formulas

Double angle formulas are a subset of trigonometric identities that provide a method to express functions involving the angle \r\(2A\) in terms of trigonometric functions involving the angle \r\(A\). These play a significant role when dealing with polynomial trigonometric equations, harmonic motion, and even in calculus.

The double angle formulas include identities for \r\(\sin(2A)\), \r\(\cos(2A)\), and \r\(\tan(2A)\). For instance, the formula for \r\(\cos(2A)\) can be \r\(\cos^2(A)-\sin^2(A)\), \r\(2\cos^2(A)-1\), or \r\(1-2\sin^2(A)\) depending on which form is most useful for the situation at hand. In our exercise, \r\(\cos(2A)\) plays a crucial role in proving the given identity by allowing the expression \r\(\cos^2(A)-\sin^2(A)\) to be compacted into \r\(\cos(2A)\), simplifying the entire equation significantly.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions to simpler forms using identities and algebraic manipulation. It's a skill that necessitates a mix of algebraic fluency and a robust understanding of trigonometric relationships. This process helps in solving equations, proving identities, and even in integration and differentiation problems in calculus.

As demonstrated in the step-by-step solution, simplification involves several stages, including the substitution of reciprocal identities, the combining of fractions, and utilizing double angle formulas. The key is recognizing which identities can be applied to transform the expression into a known form or, in some cases, proving that two expressions are equivalent, as in the exercise provided. The ultimate goal of this process is to reveal the inherent simplicity of trigonometric expressions and make them easily manageable.