Question

The value of $\cos^{-1}\left(\cos\frac{5\pi}{9}\cos\frac{\pi}{9}+\sin\frac{\pi}{9}\sin\frac{5\pi}{9}\right)$ is equal to

• A. $\frac{4\pi}{9}$
• B. $\frac{\pi}{6}$
• C. $\frac{5\pi}{9}$
• D. $\frac{\pi}{9}$
• E. $\frac{7\pi}{9}$

Hint:

Math Tip: Always double-check if your final angle lies within the principal range of the inverse trigonometric function before canceling them out. If it falls outside, you must use reference angles to bring it back into the principal range $[0, \pi]$ for $\cos^{-1}$.

Answer 1

The Correct Option is A

Concept:
Trigonometry - Compound Angle Formula for Cosine.
The fundamental identity is: $\cos(A - B) = \cos A \cos B + \sin A \sin B$.
Step 1: Identify the components of the identity.
Let's assign the given angles to the variables in our identity:

• $A = \frac{5\pi}{9}$
• $B = \frac{\pi}{9}$

Step 2: Apply the compound angle formula.
The inner expression perfectly matches the right side of the identity: $$ \cos\left(\frac{5\pi}{9}\right)\cos\left(\frac{\pi}{9}\right) + \sin\left(\frac{5\pi}{9}\right)\sin\left(\frac{\pi}{9}\right) = \cos\left(\frac{5\pi}{9} - \frac{\pi}{9}\right) $$
Step 3: Simplify the resulting angle.
Subtract the fractions: $$ \frac{5\pi}{9} - \frac{\pi}{9} = \frac{4\pi}{9} $$ So, the entire argument simplifies to $\cos\left(\frac{4\pi}{9}\right)$.
Step 4: Evaluate the inverse cosine function.
Substitute the simplified term back into the original expression: $$ \cos^{-1}\left(\cos\left(\frac{4\pi}{9}\right)\right) $$ The principal value range (domain of the output) for $\cos^{-1}(x)$ is $[0, \pi]$. Since $\frac{4\pi}{9}$ strictly falls within $[0, \pi]$, we can apply the property $\cos^{-1}(\cos \theta) = \theta$: $$ = \frac{4\pi}{9} $$