Question

If $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$, then $\cos^{2}48^{\circ}-\sin^{2}12^{\circ}$ has the value

• A. $\frac{-\sqrt{5}+1}{8}$
• B. $\frac{\sqrt{5}-1}{8}$
• C. $\frac{\sqrt{5}+1}{8}$
• D. $\frac{-1-\sqrt{5{8}$

Hint:

Logic Tip: The value of $\cos 36^\circ$ is a standard trigonometric constant, $\cos 36^\circ = \frac{\sqrt{5}+1}{4}$. If you have this memorized along with $\sin 18^\circ$, you can bypass Step 3 completely and instantly calculate $\frac{1}{2} \times \frac{\sqrt{5}+1}{4} = \frac{\sqrt{5}+1}{8}$.

Answer 1

The Correct Option is C

Concept:
The expression is in the form $\cos^2 A - \sin^2 B$, which is a standard trigonometric identity: $$\cos^2 A - \sin^2 B = \cos(A + B) \cdot \cos(A - B)$$ Additionally, we will need the double angle identity $\cos 2\theta = 1 - 2\sin^2 \theta$ to utilize the given value of $\sin 18^\circ$.
Step 1: Apply the difference of squares identity.
Let $A = 48^\circ$ and $B = 12^\circ$. Substitute these into the identity: $$\cos^2 48^\circ - \sin^2 12^\circ = \cos(48^\circ + 12^\circ) \cdot \cos(48^\circ - 12^\circ)$$ $$\cos^2 48^\circ - \sin^2 12^\circ = \cos(60^\circ) \cdot \cos(36^\circ)$$
Step 2: Substitute standard values and simplify.
We know that $\cos 60^\circ = \frac{1}{2}$. To find $\cos 36^\circ$, we can use the double angle formula in terms of sine, $\cos 2\theta = 1 - 2\sin^2 \theta$, where $\theta = 18^\circ$: $$\cos 36^\circ = 1 - 2\sin^2 18^\circ$$ Now, substitute this back into our expression: $$\text{Expression} = \frac{1}{2} \cdot \left[ 1 - 2\sin^2 18^\circ \right]$$
Step 3: Substitute the given value of $\sin 18^\circ$.
We are given $\sin 18^\circ = \frac{\sqrt{5}-1}{4}$. Square this value: $$\sin^2 18^\circ = \left( \frac{\sqrt{5}-1}{4} \right)^2 = \frac{5 + 1 - 2\sqrt{5{16} = \frac{6 - 2\sqrt{5{16} = \frac{3 - \sqrt{5{8}$$ Substitute $\sin^2 18^\circ$ into the expression: $$\text{Expression} = \frac{1}{2} \left[ 1 - 2\left(\frac{3 - \sqrt{5{8}\right) \right]$$ $$\text{Expression} = \frac{1}{2} \left[ 1 - \frac{3 - \sqrt{5{4} \right]$$ $$\text{Expression} = \frac{1}{2} \left[ \frac{4 - (3 - \sqrt{5})}{4} \right]$$ $$\text{Expression} = \frac{1}{2} \left[ \frac{1 + \sqrt{5{4} \right] = \frac{\sqrt{5} + 1}{8}$$