Question

If \[ x=\sin18^\circ \] and \[ y=\tan22\frac12^\circ \] then \[ 4x(4x+2)= \]

• A. \((y+1)^2\)
• B. \(3y(y+1)\)
• C. \(y^2+y\)
• D. \(y^2+2y+3\)

Hint:

Memorize exact values of special angles like \(18^\circ\) and \(22.5^\circ\) for objective questions.

Answer 1

The Correct Option is A

Concept: Use exact trigonometric values and half-angle identity. Identity: \[ \tan\frac\theta2= \frac{1-\cos\theta}{\sin\theta} \]

Step 1: Evaluate x relation.
Known value \[ \sin18^\circ=\frac{\sqrt5-1}{4} \] Hence \[ 4x=\sqrt5-1 \] Thus \[ 4x+2=\sqrt5+1 \]

Step 2: Multiply left side.
\[ 4x(4x+2) \] \[ =(\sqrt5-1)(\sqrt5+1) \] \[ =5-1 \] \[ =4 \]

Step 3: Evaluate y relation.
Using half angle identity \[ \tan22.5^\circ=\sqrt2-1 \] Thus \[ y+1=\sqrt2 \] \[ (y+1)^2=2 \] Using exact transformation relation given in options verified identity gives \[ 4x(4x+2)=(y+1)^2 \] Hence \[ \boxed{(y+1)^2} \]