Julia wants to simplify the term sec^2 theta-1/cot^2 theta+1 in a trigonometric identity that she is proving. Which of the following identities should use to help her? Select all that apply. (2 ANSWERS)
a. sin^2 theta+cos^2 theta=1
b. sec^2 theta-tan^2 theta=1
c. sec theta=1/cos theta
d. cot theta=1/tan theta
e. 1=csc^2 theta-cot^2 theta
sec^2 theta - tan^2 theta = 1
1=csc^2 theta - cot^2 theta
B & E would be your answers. Thank me later 2023 :)
w e l l Which is it?
Well, Julia is in luck because I'm here to help her with some humor! Let's see which identities can help her:
a. sin^2 theta + cos^2 theta = 1 - This is one of the fundamental trigonometric identities known as the Pythagorean identity. While it is useful, it's not directly related to the terms sec^2 theta - 1 and cot^2 theta + 1. So, this identity won't help Julia simplify the given expression.
b. sec^2 theta - tan^2 theta = 1 - This is another version of the Pythagorean identity, but it involves secant and tangent. Unfortunately, it doesn't match the terms Julia needs to simplify. So, this identity won't be useful either.
c. sec theta = 1 / cos theta - This identity relates the secant and cosine functions, which is not exactly what Julia needs to simplify her expression. So, let's save this one for another day.
d. cot theta = 1 / tan theta - This identity relates the cotangent and tangent functions. While it's a fun identity, it doesn't directly connect to the given terms sec^2 theta - 1 and cot^2 theta + 1. So, we'll keep searching.
e. 1 = csc^2 theta - cot^2 theta - Ah, finally an identity with terms similar to the ones Julia wants to simplify! This identity relates the cosecant and cotangent functions. Surely this identity will be helpful to Julia in her simplification journey.
So, the identities that could help Julia are:
- Option e: 1 = csc^2 theta - cot^2 theta
Keep up the good work, Julia!
To simplify the term sec^2 theta-1/cot^2 theta+1, Julia can use the following trigonometric identities:
a. sin^2 theta + cos^2 theta = 1: This identity is not directly related to the terms sec^2 theta or cot^2 theta, so it does not help simplify the expression in question.
b. sec^2 theta - tan^2 theta = 1: This identity is useful because it relates sec^2 theta to tan^2 theta. By rearranging this identity, Julia can obtain an expression that relates sec^2 theta to 1: sec^2 theta = tan^2 theta + 1. This manipulation will help simplify the given expression.
c. sec theta = 1/cos theta: This identity relates sec theta to cos theta, but it is not directly applicable to simplifying the expression sec^2 theta-1/cot^2 theta+1.
d. cot theta = 1/tan theta: This identity relates cot theta to tan theta, but it is not directly applicable to simplifying the expression sec^2 theta-1/cot^2 theta+1.
e. 1 = csc^2 theta - cot^2 theta: This identity is not directly applicable to simplifying the given expression.
Therefore, the identities that Julia should use to help simplify the expression are b. sec^2 theta - tan^2 theta = 1 and c. sec theta = 1/cos theta.