Problem 122 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \sqrt{\cos ^{2} x+\frac{1}{2}}+\sqrt{\sin ^{2} x+\frac{1}{2}}=2 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0, \pi, 2\pi, ...\)

Step 1: Square the Whole Equation

Start by squaring both sides of the equation to eliminate the square roots. This gets us: \((\cos^2(x) + \frac{1}{2}) + 2\cdot \sqrt{(\cos^2(x) + \frac{1}{2})(\sin^2(x) + \frac{1}{2})} + (\sin^2(x) + \frac{1}{2}) = 4.\)

Step 2: Simplify the Equation

Next, simplify the equation to get: \(1 + 1 + 2\cdot\sqrt{(\cos^2(x) + \frac{1}{2})(\sin^2(x) + \frac{1}{2})} = 4.\) Or, simplifying further: \(2\sqrt{(\cos^2(x) + \frac{1}{2})(\sin^2(x) + \frac{1}{2})} = 2.\)

Step 3: Isolate the Square Root

From this, isolate the square root on one side to get: \(\sqrt{(\cos^2(x) + \frac{1}{2})(\sin^2(x) + \frac{1}{2})} = 1.\)

Step 4: Square again

Square both sides of the equation again to get: \((\cos^2(x) + \frac{1}{2}) * (\sin^2(x) + \frac{1}{2}) = 1.\)

Step 5: Apply the Pythagorean identity

Now use the Pythagorean identity (\(sin^2(x) + cos^2(x) = 1\)) to replace \(\cos^2(x)\) with \(1 - \sin^2(x)\). The equation becomes: \((1-\sin^2(x) + \frac{1}{2})(\sin^2(x) + \frac{1}{2}) = 1\).

Step 6: Solve for sin^2(x)

Multiply the terms to get: \(\sin^4(x)+\frac{1}{2}-\frac{1}{2}\sin^2(x)=0.\) This is a quadratic equation in \(\sin^2(x)\). Solve it as a quadratic to get \(\sin^2(x) = 0\) or \(\sin^2(x) = 1\). The solutions for \(x\) are then \(x = 0, \pi, 2\pi, ...\)

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry and relates the squares of sine and cosine functions of an angle. It states that for any angle \(x\): \[\sin^2x + \cos^2x = 1\] This identity is derived from the Pythagorean theorem applied to a unit circle.

• Unit Circle: A circle centered at the origin with a radius of 1.
• Any point on the circle can be represented by \((\cos x, \sin x)\).
• Therefore, by the Pythagorean theorem, the relationship holds true.

In solving trigonometric equations like the one given, the Pythagorean Identity helps simplify expressions by converting squares of sine or cosine into one another. Understanding this identity allows for quick and efficient simplification and solution of trigonometric problems.

Quadratic Equation

The quadratic equation is a powerful mathematical tool that takes the form: \[ax^2 + bx + c = 0\] In the context of trigonometric equations, the variable might instead be some trigonometric function like \(\sin^2(x)\) or \(\cos^2(x)\). Here's how it was used in the problem:

• The equation from step 6 becomes \(\sin^4(x) + \frac{1}{2} - \frac{1}{2}\sin^2(x) = 0\).
• Consider \(\sin^2(x)\) as a single variable \(y\), transforming it into \(y^2 - \frac{1}{2}y + \frac{1}{2} = 0\).

To solve, use the quadratic formula: \[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Identifying \(a = 1\), \(b = -\frac{1}{2}\), and \(c = \frac{1}{2}\), you find solutions for \(y\), which correspond back to \(\sin^2(x)\). This allows you to determine the values of \(x\) that satisfy the original equation.

Square Roots Elimination

Square root elimination is crucial for simplifying equations involving square roots. It helps by transforming a sometimes complex equation into a more manageable quadratic one. Here’s a simple approach:

• Square both sides of an equation where roots are present. For example, if \(\sqrt{a} = b\), then squaring both sides gives \(a = b^2\).
• In the original problem, squaring was applied to get rid of the roots, eventually simplifying down to a product term that could be squared again if needed.

When the square roots are paired with trigonometric terms, handling them carefully ensures that further simplifications remain valid. This technique is often the first step in reducing the complexity of equations, setting the stage for other algebraic manipulations to resolve the problem.

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that hold true for any angle. They don't just include the Pythagorean Identity but also encompass other key identities:

• Reciprocal Identities: \(\sin x = 1/\csc x\), \(\cos x = 1/\sec x\), \(\tan x = 1/\cot x\).
• Angle Sum and Difference: \(\sin(a \pm b)\), \(\cos(a \pm b)\).
• Double Angle: \(\sin(2x) = 2\sin x\cos x\), \(\cos(2x) = \cos^2x - \sin^2x\).

In solving equations, these identities allow simplification and transformation of trig expressions. They ensure that the equations can be rewritten in forms that are easier to solve. Recognizing which identity to apply comes with practice, enhancing problem-solving speed and accuracy.