Question

Value of \(\cos ^{2} 48^{\circ}-\sin ^{2} 12^{\circ}\) (a) \(\frac{\sqrt{5}+1}{4}\) (b) \(\frac{\sqrt{5}+1}{8}\) (c) \(\frac{\sqrt{5}-1}{8}\) (d) \(\frac{\sqrt{5}-1}{4}\)

Short answer

Answer: \(\frac{\sqrt{5}-1}{2}\).

Step-by-step solution

Rewrite the expression using angle relationship

Rewrite the given expression using the fact that \(\cos(90^\circ - \theta) = \sin(\theta)\): \(\cos ^{2} 48^{\circ}-\sin ^{2} 12^{\circ} = \cos^2(90^\circ - 48^\circ) - \sin^2(12^\circ)\) \(=\cos^2(42^\circ) - \sin^2(12^\circ)\)

Apply the angle subtraction formula

Using the identity \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\), we can rewrite the expression as: \(\cos^2(42^\circ) - \sin^2(12^\circ) = \cos(2 \cdot 12^\circ) = \cos(24^\circ)\)

Apply the double-angle formula

Now use the double-angle formula for cosine: \(\cos(24^\circ) = 2\cos^2(12^\circ) - 1\)

Solve for the desired expression

From Step 2, we want to find the value of \( \cos^2(42^\circ) - \sin^2(12^\circ) = \cos(24^\circ)\). So we can just solve the equation from Step 3: \(\cos(24^\circ) = 2\cos^2(12^\circ) - 1\) Now we know that \(\cos(48^\circ)=\cos(24^\circ+24^\circ)\), from which we can use the sum-to-product formula, \(\cos(24^\circ+24^\circ) = \cos(24^\circ)\cos(24^\circ) - \sin(24^\circ)\sin(24^\circ)\). Then note that \(\sin(24^\circ)=\cos(66^\circ)=\cos(90^\circ-24^\circ)\), and then use the sum-to-product formula again, \(\sin(24^\circ)=\cos(66^\circ) = \cos(24^\circ)\cos(42^\circ) - \sin(24^\circ)\sin(42^\circ)\). Square the last equation and rearrange to get, \([\sin^2(24^\circ)][\sin^2(42^\circ)]=[\cos^2(24^\circ)][\cos^2(42^\circ)-1]\). Use the relationship \(\sin^2(\theta)=1-\cos^2(\theta)\) to substitute, then we have, \((1-\cos^2(24^\circ))(1-\cos^2(42^\circ))= \cos^2(24^\circ)[\cos^2(42^\circ) - 1]\). Now substitute the equation from Step 3, \(\cos(24^\circ)=2\cos^2(12^\circ)-1\), we get, \((1-(2\cos^2(12^\circ)-1))(1-\cos^2(42^\circ))=(2\cos^2(12^\circ)-1)[\cos^2(42^\circ)-1]\). Simplify the equation, we get, \(1-3\cos^2(12^\circ)+2\cos^4(12^\circ)=0\). Let \(x=\cos^2(12^\circ)\), then the above equation becomes, \(2x^2-3x+1=0\), which has solutions \(\frac{1\pm\sqrt{5}}{4}\). Since \(0\le x = \cos^2(12^\circ)\le 1\), we have \(x=\frac{1+\sqrt{5}}{4}\). Now we can find the desired value, \(\cos(24^\circ)=2\cos^2(12^\circ)-1=2(\frac{1+\sqrt{5}}{4})-1=\frac{\sqrt{5}+1}{2}-1=\frac{\sqrt{5}-1}{2}\). Thus, our answer is \(\cos^2(42^\circ)-\sin^2(12^\circ)=\cos(24^\circ)=\frac{\sqrt{5}-1}{2}\), which matches with option (d).

Key concepts

Cosine and Sine Relationships

Trigonometric identities are mathematical tools that help us establish relationships between different trigonometric functions. One of the core relationships is between cosine and sine, represented by the identity:

• \( \cos(90^\circ - \theta) = \sin(\theta) \)

This equation is known as a co-function identity. It showcases how the sine and cosine functions complement each other, particularly when dealing with complementary angles.
For example, in this problem, the cosine of 42 degrees is the same as the sine of 48 degrees because 42 and 48 are complementary with respect to 90 degrees.
Understanding these relationships allows you to simplify expressions, solve equations, and better comprehend the harmonic nature of trigonometric functions.

Double-Angle Formula

The double-angle formulas are crucial identities in trigonometry that allow for the simplification of expressions involving trigonometric functions of double angles. One of the most commonly used double-angle formulas is for cosine:

• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

This identity is particularly useful when you need to express a trigonometric function of a double angle in terms of single-angle functions.
In our original problem, this formula helps simplify the expression \( \cos^2(42^\circ) - \sin^2(12^\circ) \) into a form that relates directly to a single angle (24 degrees).
By applying the double-angle formula, complex trigonometric expressions can be simplified into more manageable equations, thus making the solving process more straightforward.

Trigonometric Equations

Trigonometric equations are mathematical equations that involve trigonometric functions like sine, cosine, and tangent. Solving these equations often requires applying several trigonometric identities and manipulations.
In this exercise, after employing trigonometric identities like co-function and double-angle formulas, we reduced a complex expression into a simpler trigonometric equation. Solving for \( x = \cos^2(12^\circ) \) showed how algebra and trigonometry can intertwine.
This part of the problem highlights the following steps in handling trigonometric equations:

• Use identities to simplify complex expressions
• Transform them into polynomial equations when possible
• Solve the polynomial using algebraic methods or online calculators

These techniques allow us to employ a range of mathematical skills, making trigonometry a versatile part of mathematics.