Find the exact value of the expression. Do not use a calculator.
-$1 - \tan ^ { 2 } 35 ^ { \circ } + \csc ^ { 2 } 55 ^ { \circ }$
A) 2
B) 0
C) $- 1$
D) 1 )1

Question

Quizplus · Accepted Answer

Using trigonometric identities, we can simplify the expression. The identity $1 + \tan^2 \theta = \sec^2 \theta$ can be rearranged to $1 - \tan^2 \theta = -\tan^2 \theta + 1 = \cos^2 \theta - \sin^2 \theta$, which is a form of the cosine double angle formula. However, the key to solving this problem lies in recognizing the complementary angle relationships: $\tan(35^\circ) = \cot(55^\circ)$ and $\csc(55^\circ) = \sec(35^\circ)$. Thus, the expression simplifies using $\tan^2(35^\circ) + 1 = \sec^2(35^\circ)$, leading to $1 - \tan^2(35^\circ) + \csc^2(55^\circ) = 1 + 1 = 2$, because $\csc^2(55^\circ) = \sec^2(35^\circ)$.

Find the Exact Value of the Expression 1−tan⁡235∘+csc⁡255∘1 - \tan ^ { 2 } 35 ^ { \circ } + \csc ^ { 2 } 55 ^ { \circ }1−tan235∘+csc255∘

Find the Exact Value of the Expression $1 - \tan ^ { 2 } 35 ^ { \circ } + \csc ^ { 2 } 55 ^ { \circ }$