Use the cofunction identities to evaluate the expression below without the aid of a calculator.
$\cos ^ { 2 } 50 ^ { \circ } + \cos ^ { 2 } 58 ^ { \circ } + \cos ^ { 2 } 40 ^ { \circ } + \cos ^ { 2 } 32 ^ { \circ }$
A) 1
B) 2
C) $- 1$
D) 0
E) $\frac { 1 } { 2 }$

Question

Quizplus · Accepted Answer

The cofunction identities state that $\sin(x) = \cos(90^\circ - x)$ and $\cos(x) = \sin(90^\circ - x)$. Using these identities, we can pair the given cosines as follows: $\cos^2(50^\circ)$ with $\cos^2(40^\circ)$ and $\cos^2(58^\circ)$ with $\cos^2(32^\circ)$, because $50^\circ + 40^\circ = 90^\circ$ and $58^\circ + 32^\circ = 90^\circ$. This means each pair sums to 1 since $\sin^2(x) + \cos^2(x) = 1$. Therefore, the total sum is $1 + 1 = 2$.

Use the Cofunction Identities to Evaluate the Expression Below Without cos⁡250∘+cos⁡258∘+cos⁡240∘+cos⁡232∘\cos ^ { 2 } 50 ^ { \circ } + \cos ^ { 2 } 58 ^ { \circ } + \cos ^ { 2 } 40 ^ { \circ } + \cos ^ { 2 } 32 ^ { \circ }cos250∘+cos258∘+cos240∘+cos232∘

Use the Cofunction Identities to Evaluate the Expression Below Without $\cos ^ { 2 } 50 ^ { \circ } + \cos ^ { 2 } 58 ^ { \circ } + \cos ^ { 2 } 40 ^ { \circ } + \cos ^ { 2 } 32 ^ { \circ }$