[tex]\huge✿\mathcal{\fcolorbox{grey}{black}{{{\red{answer}}}}}✿[/tex]

To find the value of sin^4(10°) + sin^4(50°) + sin^4(70°), we can use the trigonometric identity:

sin^2(x) + cos^2(x) = 1

Rearranging the equation, we get:

sin^2(x) = 1 - cos^2(x)

Now, let's substitute this into the expression:

sin^4(10°) + sin^4(50°) + sin^4(70°) = [1 - cos^2(10°)]^2 + [1 - cos^2(50°)]^2 + [1 - cos^2(70°)]^2

Using the identity sin^2(x) = 1 - cos^2(x), we can simplify further:

sin^4(10°) + sin^4(50°) + sin^4(70°) = [1 - cos^2(10°)]^2 + [1 - cos^2(50°)]^2 + [1 - cos^2(70°)]^2

= (1 - cos^2(10°))^2 + (1 - cos^2(50°))^2 + (1 - cos^2(70°))^2

Expanding and simplifying each term:

sin^4(10°) + sin^4(50°) + sin^4(70°) = (1 - 2cos^2(10°) + cos^4(10°)) + (1 - 2cos^2(50°) + cos^4(50°)) + (1 - 2cos^2(70°) + cos^4(70°))

= 3 - 2(cos^2(10°) + cos^2(50°) + cos^2(70°)) + (cos^4(10°) + cos^4(50°) + cos^4(70°))

Now, we can use the identity cos^2(x) + sin^2(x) = 1 to simplify further:

cos^2(x) = 1 - sin^2(x)

Plugging this into the equation:

sin^4(10°) + sin^4(50°) + sin^4(70°) = 3 - 2[(1 - sin^2(10°)) + (1 - sin^2(50°)) + (1 - sin^2(70°))] + [(1 - sin^2(10°))^2 + (1 - sin^2(50°))^2 + (1 - sin^2(70°))^2]

= 3 - 2[3 - (sin^2(10°) + sin^2(50°) + sin^2(70°))] + [(1 - sin^2(10°))^2 + (1 - sin^2(50°))^2 + (1 - sin^2(70°))^2]

We know that sin^2(x) + cos^2(x) = 1, so:

sin^2(x) = 1 - cos^2(x)

Plugging this into the equation again:

sin^4(10°) + sin^4(50°) + sin^4(70°) = 3 - 2[3 - (1 - cos^2(10°) + 1 - cos^2(50°) + 1 - cos^2(70°))] + [(1 - sin^2(10°))^2 + (1 - sin^2(50°))^2 +

(1 - sin^2(70°))^2]

= 3 - 2[3 - (3 - cos^2(10°) - cos^2(50°) - cos^2(70°))] + [(1 - sin^2(10°))^2 + (1 - sin^2(50°))^2 + (1 - sin^2(70°))^2]

= 3 - 2[3 - (3 - cos^2(10°) - cos^2(50°) - cos^2(70°))] + [(1 - (1 - cos^2(10°)))^2 + (1 - (1 - cos^2(50°)))^2 + (1 - (1 - cos^2(70°)))^2]

= 3 - 2[3 - (3 - cos^2(10°) - cos^2(50°) - cos^2(70°))] + [(cos^2(10°))^2 + (cos^2(50°))^2 + (cos^2(70°))^2]

Now, we can calculate the values of cos^2(10°), cos^2(50°), and cos^2(70°):

cos(10°) ≈ 0.9848

cos(50°) ≈ 0.6428

cos(70°) ≈ 0.3420

cos^2(10°) ≈ (0.9848)^2 ≈ 0.9696

cos^2(50°) ≈ (0.6428)^2 ≈ 0.4137

cos^2(70°) ≈ (0.3420)^2 ≈ 0.1169

Plugging these values back into the equation:

sin^4(10°) + sin^4(50°) + sin^4(70°) = 3 - 2[3 - (3 - 0.9696 - 0.4137 - 0.1169)] + [0.0304^2 + 0.5863^2 + 0.8831^2]

= 3 - 2[3 - (2.0034)] + [0.0009 + 0.3435 + 0.7799]

= 3 - 2[1.9966] + 1.1243

= 3 - 3.9932 + 1.1243

= 0.1311

Therefore, the value of sin^4(10°) + sin^4(50°) + sin^4(70°) is approximately 0.1311.

The closest option to this value is option D: 9/4.