Question:

Step 1: Simplify the numerator using complementary angle identity. The given expression is $\frac{\cos 68^\circ}{\sin 56^\circ \sin 34^\circ \tan 22^\circ}$. We use the complementary angle identity $\cos \theta = \sin (90^\circ - \theta)$. Since $68^\circ + 22^\circ = 90^\circ$, we can rewrite $\cos 68^\circ$ as $\sin(90^\circ - 68^\circ) = \sin 22^\circ$. Step 2: Simplify the product of sine terms in the denominator using the product-to-sum identity. The product-to-sum identity for $\sin A \sin B$ is $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$. For $\sin 56^\circ \sin 34^\circ$: Let $A = 56^\circ$ and $B = 34^\circ$. Then $A-B = 56^\circ - 34^\circ = 22^\circ$. And $A+B = 56^\circ + 34^\circ = 90^\circ$. Substituting these values: $$ \sin 56^\circ \sin 34^\circ = \frac{1}{2}[\cos 22^\circ - \cos 90^\circ] $$ Since $\cos 90^\circ = 0$: $$ \sin 56^\circ \sin 34^\circ = \frac{1}{2}[\cos 22^\circ - 0] = \frac{1}{2}\cos 22^\circ $$ Step 3: Substitute the simplified numerator and the product of sine terms back into the original expression. The original expression becomes: $$ \frac{\sin 22^\circ}{\left(\frac{1}{2}\cos 22^\circ\right) \tan 22^\circ} $$ Step 4: Express $\tan 22^\circ$ in terms of sine and cosine, and simplify the expression. We use the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$. Substitute $\tan 22^\circ = \frac{\sin 22^\circ}{\cos 22^\circ}$ into the expression: $$ \frac{\sin 22^\circ}{\frac{1}{2}\cos 22^\circ \cdot \frac{\sin 22^\circ}{\cos 22^\circ}} $$ Now, cancel out $\cos 22^\circ$ in the denominator: $$ \frac{\sin 22^\circ}{\frac{1}{2}\sin 22^\circ} $$ Finally, cancel out $\sin 22^\circ$ from the numerator and denominator: $$ \frac{1}{\frac{1}{2}} = 2 $$ Step 5: Conclude the calculation. The value of the given expression is $2$. The provided original solution contains conflicting statements regarding the final answer. The explicit calculations within the original solution consistently lead to $2$. The assertion of $16$ in the original solution is not supported by the provided mathematical steps. Therefore, based on the logical steps and calculations given, the value is $2$. None of the provided options (P) 16 or (Q) 3 match the calculated value of $2$.