Question

What is equivalent to sec ² (x)(1−cos 2 (x))?

A) 1
B) tan ²(x)
C) csc ² (x)
D) cot ² (x)

Answer 1

The expression sec ² (x)(1−cos 2 (x)) simplifies down to tan ²(x) by applying trigonometric identities, specifically the double angle identity for cosine and the definition for tangent.

Step-by-step explanation:

The expression equivalent to sec ² (x)(1−cos 2 (x)) can be determined by using trigonometric identities. The double angle identity for cosine states that cos 2x = 2cos²x − 1, which can also be written as cos 2x = 1 − 2sin²x. Using this identity, we can rewrite the expression as sec²x · 2sin²x. Remember that sec x = 1/cos x and since sin²x = 1 − cos²x, the expression becomes (1/cos²x) · 2(1 − cos²x), which simplifies to 2 sin²x. The trigonometric identity tan²x = sin²x/cos²x then allows us to express 2 sin²x as 2 tan²x, upon which we can see that the 2 cancels and the answer simplifies to tan²x.

Therefore, the expression sec ² (x)(1−cos 2 (x)) is equivalent to option B) tan ²(x).