Use the cofunction identities to evaluate the expression without using a calculator. $\tan ^ { 2 } 71 + \cot ^ { 2 } 18 - \sec ^ { 2 } 72 - \csc ^ { 2 } 19$
A) $- 1$
B) 3
C) 1
D) $- 2$
E) 2

Question

Use the cofunction identities to evaluate the expression without using a calculator. $\tan ^ { 2 } 71 + \cot ^ { 2 } 18 - \sec ^ { 2 } 72 - \csc ^ { 2 } 19$ 
A) $- 1$
B) 3
C) 1
D) $- 2$
E) 2

Quizplus · Accepted Answer

Using cofunction identities: $\tan(90^\circ - \theta) = \cot(\theta)$, $\sec(90^\circ - \theta) = \csc(\theta)$. Thus, $\tan^2 71 = \cot^2 19$ and $\sec^2 72 = \csc^2 18$. The expression simplifies to $\cot^2 19 + \cot^2 18 - \csc^2 18 - \csc^2 19$. Using $\cot^2 \theta = \csc^2 \theta - 1$, it becomes $(\csc^2 19 - 1) + (\csc^2 18 - 1) - \csc^2 18 - \csc^2 19 = -2$.

Use the Cofunction Identities to Evaluate the Expression Without Using tan⁡271+cot⁡218−sec⁡272−csc⁡219\tan ^ { 2 } 71 + \cot ^ { 2 } 18 - \sec ^ { 2 } 72 - \csc ^ { 2 } 19tan271+cot218−sec272−csc219

Use the Cofunction Identities to Evaluate the Expression Without Using $\tan ^ { 2 } 71 + \cot ^ { 2 } 18 - \sec ^ { 2 } 72 - \csc ^ { 2 } 19$