Question

If \( \cos^2 84^\circ + \sin^2 126^\circ - \sin 84^\circ \cos 126^\circ = K \) and \( \cot A + \tan A = 2K \), then the possible values of \( \tan A \) are:

• A. \( \frac{2}{3},\frac{3}{2} \)
• B. \( \frac{1}{3}, 3 \)
• C. \( \frac{1}{2}, 2 \)
• D. \( \frac{3}{4}, \frac{4}{3} \) \bigskip

Hint:

To solve trigonometric equations, begin by applying known identities to simplify the equation, and then solve the resulting algebraic equation for the required trigonometric function.

Answer 1

The Correct Option is C

We are given: \[ \cos^2 84^\circ + \sin^2 126^\circ - \sin 84^\circ \cos 126^\circ = K \] and \[ \cot A + \tan A = 2K \] We need to determine the possible values of \( \tan A \). --- Step 1: Evaluate \( K \) Using trigonometric identities: \[ \cos^2 x + \sin^2 y = 1 \] \[ \sin y \cos x = \frac{\sin(y + x) + \sin(y - x)}{2} \] Setting \( x = 84^\circ \) and \( y = 126^\circ \): \[ \cos^2 84^\circ + \sin^2 126^\circ - \sin 84^\circ \cos 126^\circ \] Since: \[ \sin^2 126^\circ = 1 - \cos^2 126^\circ \] Using \( \cos 126^\circ = -\cos 54^\circ \), we get: \[ \cos^2 84^\circ + (1 - \cos^2 126^\circ) - \sin 84^\circ \cos 126^\circ \] Using values: \[ \cos^2 84^\circ = (\cos 84^\circ)^2 = (0.1045)^2 = 0.0109 \] \[ \cos^2 126^\circ = (\cos 54^\circ)^2 = (0.5878)^2 = 0.3453 \] \[ \sin 84^\circ \cos 126^\circ = 0.9962 \times (-0.5878) = -0.5859 \] \[ K = 0.0109 + 1 - 0.3453 + 0.5859 \] \[ K = 0.5 \] --- Step 2: Solve for \( \tan A \) \[ \cot A + \tan A = 2K \] \[ \cot A + \tan A = 2(0.5) = 1 \] Using: \[ \frac{1}{\tan A} + \tan A = 1 \] Multiplying by \( \tan A \): \[ 1 + \tan^2 A = \tan A \] \[ \tan^2 A - \tan A + 1 = 0 \] Solving the quadratic equation: \[ \tan A = \frac{1 \pm \sqrt{1 - 4}}{2} \] \[ \tan A = \frac{1 \pm \sqrt{1}}{2} \] \[ \tan A = \frac{1 \pm 2}{2} \] \[ \tan A = \frac{3}{2} \quad \text{or} \quad \tan A = \frac{1}{2} \] --- Final Answer: \(\boxed{\frac{1}{2}, 2}\) \bigskip