Use the cofunction identities to evaluate the expression below
sin^2 21° + sin^2 61° + sin^2 69° + sin^2 29°
2
sin(21º) = cos(69º)
sin(29º) = cos(61º)
so it's twice sin² + cos²
To evaluate the given expression using cofunction identities, we need to rewrite each sine squared term in terms of its corresponding cofunction identity, which is cosine.
The cofunction identities state that the sine of an angle is equal to the cosine of its complementary angle, and the cosine of an angle is equal to the sine of its complementary angle.
Let's find the complementary angles for each of the given angles:
Complementary angle of 21°: 90° - 21° = 69°
Complementary angle of 61°: 90° - 61° = 29°
Complementary angle of 69°: 90° - 69° = 21°
Complementary angle of 29°: 90° - 29° = 61°
Now we can rewrite each sine squared term using the cofunction identities:
sin^2 21° = cos^2 69°
sin^2 61° = cos^2 29°
sin^2 69° = cos^2 21°
sin^2 29° = cos^2 61°
Substituting these rewritten terms into the original expression:
cos^2 69° + cos^2 29° + cos^2 21° + cos^2 61°
Now, we can use a trigonometric identity that states that the sum of the squares of the cosine and sine of any angle is always equal to 1.
cos^2 x + sin^2 x = 1
Using this identity, we can simplify the expression:
cos^2 69° + cos^2 29° + cos^2 21° + cos^2 61° = 1 + 1 + 1 + 1 = 4
Therefore, the value of the given expression is 4.