Simplify
. a. 2 b.
c.
d. 1 Please select the best answer from the choices provided A B C D
The Correct Answer and Explanation is:
We are asked to simplify the expression: 2(csc2θ−cot2θ)2(\csc^2 \theta – \cot^2 \theta)
Step 1: Recall Trigonometric Identity
One of the Pythagorean identities in trigonometry is: csc2θ−cot2θ=1\csc^2 \theta – \cot^2 \theta = 1
Step 2: Substitute the Identity
Using the identity above: 2(csc2θ−cot2θ)=2(1)=22(\csc^2 \theta – \cot^2 \theta) = 2(1) = 2
Final Answer:
2\boxed{2}
So, the correct option is:
A. 2
Explanation:
To solve the trigonometric expression 2(csc2θ−cot2θ)2(\csc^2 \theta – \cot^2 \theta), we use one of the fundamental Pythagorean identities from trigonometry. These identities are foundational tools in simplifying trigonometric expressions and solving equations.
One such identity is: csc2θ=1+cot2θ\csc^2 \theta = 1 + \cot^2 \theta
This identity can be rearranged to isolate the expression we are working with: csc2θ−cot2θ=1\csc^2 \theta – \cot^2 \theta = 1
This tells us that no matter what the angle θ\theta is (assuming it’s within the domain of the cosecant and cotangent functions), the expression csc2θ−cot2θ\csc^2 \theta – \cot^2 \theta will always simplify to 1.
Therefore, we substitute this result into our original expression: 2(csc2θ−cot2θ)=2(1)=22(\csc^2 \theta – \cot^2 \theta) = 2(1) = 2
This type of question tests your understanding of trigonometric identities, which are essential in both basic and advanced trigonometry. It also demonstrates how recognizing patterns and identities can drastically simplify what might at first appear to be a complex expression. Understanding and memorizing these identities not only helps in simplifying expressions but also in calculus, physics, and engineering problems where trigonometric functions are frequently used.
So, the simplified value is 2, making option A the correct choice.
