The value of cos²29° + cos²61° is:

The correct answer is 1.

To find the value of the expression, we can use the trigonometric identity for complementary angles:

$\cos (90°-\theta )=\sin \theta$

Here, we can rewrite the second term, $\cos 61°$, in terms of sine by expressing $61°$ as $90°-29°$:

$\cos 61°=\cos (90°-29°)=\sin 29°$

Squaring both sides of this equation, we get:

${\cos }^{2}61°={\sin }^{2}29°$

Now, substituting this value back into the original expression gives:

${\cos }^{2}29°+{\cos }^{2}61°={\cos }^{2}29°+{\sin }^{2}29°$

Applying the standard Pythagorean trigonometric identity ${\sin }^2\theta +{\cos }^2\theta =1$ for $\theta =29°$, we have:

${\cos }^{2}29°+{\sin }^{2}29°=1$

Thus, the value of the given expression is 1.