If sec 6A = cosec (A – 29°), where 2A is an acute angle, then the measure of ∠A = ___°.

The correct answer is 17.

Step 1: Understand the trigonometric identity for complementary angles.

We know that secant and cosecant are complementary trigonometric functions, which gives the identity:

$\sec \left(\theta \right)=\mathrm{cosec}(90°-\theta )$

Using this complementary angle relationship, we can express $\sec \left(6A\right)$ in terms of cosecant:

$\sec \left(6A\right)=\mathrm{cosec}(90°-6A)$

Step 2: Set up the equation using the given expression.

The given equation is:

$\sec \left(6A\right)=\mathrm{cosec}(A-29°)$

Substituting $\mathrm{cosec}(90°-6A)$ for $\sec \left(6A\right)$:

$\mathrm{cosec}(90°-6A)=\mathrm{cosec}(A-29°)$

Step 3: Equate the angles and solve for A.

Since the cosecant values of both acute angles are equal, their angle measures must be equal:

$90°-6A=A-29°$

Group the terms containing $A$ on one side and constant values on the other:

$90°+29°=A+6A$

$119°=7A$

$A=\frac{119°}{7}$

$A=17°$

Therefore, the measure of ∠A is 17°.