Find the ratio of (sin² 22° + sin² 68°) and (cos² 33° + cos² 57°).

Correct option is B

Given:

We need to find the ratio of

$$\sin^2 22^\circ + \sin^2 68^\circ and \cos^2 33^\circ + \cos^2 57^\circ .$$​

Formula Used:

1. Use the trigonometric identity: $$\sin^2 \theta + \cos^2 \theta = 1$$​ .
2. Use complementary angles for sine and cosine:
​$$- \sin(90^\circ - \theta) = \cos \theta$$​
​$$- \cos(90^\circ - \theta) = \sin \theta$$​

Solution:

1. Calculate $$\sin^2 22^\circ + \sin^2 68^\circ$$​ :

- Using the complementary angle property, $$\sin 68^\circ = \cos 22^\circ$$​ .
- Therefore, $$\sin^2 22^\circ + \sin^2 68^\circ = \sin^2 22^\circ + \cos^2 22^\circ$$​ .

- Using $$\sin^2 \theta + \cos^2 \theta = 1$$​ , we get:
​$$\sin^2 22^\circ + \sin^2 68^\circ = 1$$​ .

2. Calculate $$\cos^2 33^\circ + \cos^2 57^\circ$$​ :

- Using the complementary angle property, $$\cos 57^\circ = \sin 33^\circ$$​ 
- Therefore, $$\cos^2 33^\circ + \cos^2 57^\circ = \cos^2 33^\circ + \sin^2 33^\circ$$​ 

- Using $$\sin^2 \theta + \cos^2 \theta = 1$$​ , we get:
​$$\cos^2 33^\circ + \cos^2 57^\circ = 1$$​ .

3. Find the Ratio:

The ratio is $$\frac{\sin^2 22^\circ + \sin^2 68^\circ}{\cos^2 33^\circ + \cos^2 57^\circ} = \frac{1}{1} = 1$$​ .

Answer: The ratio is 1:1.