Let's simplify the expression step by step:

cos^2 83 is equal to cos^2(90-7), which using the cosine double angle formula, becomes 1/2(1 + cos(2*7)), which simplifies to 1/2(1 + cos 14).

cos^2 174 is equal to cos^2(180-6), which using the cosine double angle formula again, becomes 1/2(1 + cos(2*6)), which simplifies to 1/2(1 + cos 12).

Now the expression becomes:

35 / (1/2(1 + cos 14) + 1/2(1 + cos 12))

35 / (1/2 + 1/2 + 1/2cos 14 + 1/2cos 12)

35 / (1 + 1/2cos 14 + 1/2cos 12)

At this point, we need the exact values of cos 14 and cos 12 to simplify it further.

cos 14 ≈ 0.9701
cos 12 ≈ 0.9781

Now, substitute the values into the expression:

35 / (1 + 1/20.9701 + 1/20.9781)

35 / (1 + 0.4851 + 0.48905)

35 / (1 + 0.97415)

35 / 1.97415

≈ 17.71

Therefore, 35/(cos^2 83 + cos^2 174) is approximately 17.71.