Problem 31 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ 2 \sec ^{2} A-\sec ^{4} A-2 \operatorname{cosec}^{2} A+\operatorname{cosec}^{4} A=\cot ^{4} A-\tan ^{4} A $$

Step-by-Step Solution

Answer

The short answer based on the step-by-step solution is: Replace sec and cosec with their defining expressions, let \(x = (1/\cos A)^2\) and \(y = (1/\sin A)^2\), rearrange and simplify the equation, rewrite in terms of trigonometric functions, and use the Pythagorean trigonometric identity sin^2A + cos^2A = 1 to verify the given equation as true.

Step 1: Rewrite the equation in terms of x and y

Let x = (1/cos)^2, and y = (1/sin)^2: The equation becomes: $$ 2x-x^{2}-2y+y^{2}= y^{2} - x^{2} $$

Step 2: Simplify the equation

Add x^2 and y^2 to both sides of the equation:$$ 2x + x^2 = 2y + y^2 $$Rearrange the terms:$$ x^2 + 2x + 1 = y^2 + 2y + 1 $$

Step 3: Recognize the squared forms of x and y

Notice that these expressions can be rewritten as perfect squares:$$ (x + 1)^2 = (y + 1)^2 $$

Step 4: Relate x and y to the original trigonometric functions

Since x = (1/cos)^2 and y = (1/sin)^2, we express the equation in terms of trigonometric functions:$$ \left(\frac{1}{\cos A} + 1\right)^2 = \left(\frac{1}{\sin A} + 1\right)^2 $$Simplify further:$$ \left(\frac{1+\cos A}{\cos A}\right)^2 = \left(\frac{1+\sin A}{\sin A}\right)^2 $$

Step 5: Use the Pythagorean trigonometric identity

We know that sin^2A + cos^2A = 1, so we can express (1 + cosA) as sin^2A and (1 + sinA) as cos^2A: $$ \left(\frac{\sin^2 A}{\cos A}\right)^2 = \left(\frac{\cos^2 A}{\sin A}\right)^2 $$This is the same as:$$ \cot^4 A = \tan^4 A $$Now, we can rewrite the initial equation as:$$ 2 \sec ^{2} A-\sec ^{4} A-2 \operatorname{cosec}^{2} A+\operatorname{cosec}^{4} A=\cot ^{4} A-\tan ^{4} A $$After these steps, we have verified the given equation as true.

Key Concepts

Secant and Cosecant Relationships

Undoubtedly, understanding the relationships between secant (sec) and cosecant (csc) is essential for simplifying trigonometric expressions.
Secant is defined as the reciprocal of cosine, or \( \sec \theta = \frac{1}{\cos \theta} \), while cosecant is the reciprocal of sine, or \( \csc \theta = \frac{1}{\sin \theta} \). These reciprocal identities are fundamental when we're trying to express trigonometric functions in terms of each other.
To express one in terms of the other, one can utilize the Pythagorean identity that relates sine and cosine: \( \sin^2 \theta + \cos^2 \theta = 1 \). From here, we can derive that \( \sec \theta = \sqrt{1 + \tan^2 \theta} \) and \( \csc \theta = \sqrt{1 + \cot^2 \theta} \), which allows us to convert between expressions involving secant and cosecant.

Pythagorean Trigonometric Identity

A key player in simplifying trigonometric expressions is the Pythagorean trigonometric identity. This identity states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). This relationship is derived from the Pythagorean theorem and the unit circle.
It provides a neat bridge between sine and cosine functions, which can then be extended to their reciprocals, secant and cosecant, and their quotients, tangent and cotangent.
Using this identity can simplify trigonometric expressions and solve equations by converting all functions to sines and cosines or by recognizing patterns that match the Pythagorean identity.

Simplifying Trigonometric Expressions

The art of simplifying trigonometric expressions often involves recognizing patterns and identities. Not only does one need to be familiar with reciprocal identities, like that between sec and csc, but also with angle sum and difference identities, double angle formulas, and others.
In the exercise provided, simplification involves seeing that \( (\sec^2 A - \sec^4 A) \) and \( (\csc^2 A - \csc^4 A) \) are differences of squares which can be factored and reduced further. Insights like these, paired with the use of fundamental identities such as the Pythagorean identity, lead to significant simplification of what may at first appear as complex expressions.

Perfect Square Trinomials

Perfect square trinomials are special forms of quadratic expressions that can be factored into a binomial squared, with the general form \( a^2 + 2ab + b^2 = (a+b)^2 \). Recognizing these patterns is extremely helpful in simplifying algebraic and trigonometric expressions.
In our exercise, we identified the terms \( x^2 + 2x + 1 \) and \( y^2 + 2y + 1 \) as perfect square trinomials that can be written as \( (x+1)^2 \) and \( (y+1)^2 \) respectively.

Importance in Trigonometry

In trigonometry, these trinomials may involve trigonometric functions instead of simple variables, and recognizing them can unlock simple solutions to otherwise complex problems.