Question

Prove the trigonometric identity (1-cos²A)csc²A = 1.

A. (1-sin²A)csc²A = 1
B. (1-sin²A)sec²A = 1
C. (1-tan²A)csc²A = 1
D. (1-tan²A)sec²A = 1

Answer 1

To prove the trigonometric identity (1-cos²A)csc²A = 1, we use the Pythagorean identity to express 1-cos²A as sin²A, then recognize that csc²A equals 1/sin²A, resulting in a simplification to 1.

Step-by-step explanation:

The question relates to proving a trigonometric identity. We are asked to verify whether (1-cos²A)csc²A = 1. We can use the Pythagorean identity sin²A + cos²A = 1, which can be rearranged to 1 - cos²A = sin²A. Substituting this into our original expression, we have (sin²A)csc²A.

Recall that cscA = 1/sinA; therefore, csc²A = 1/sin²A. Substituting csc²A in the expression, we get (sin²A)(1/sin²A), which simplifies to 1. Therefore, the original trigonometric identity is proven to be correct.