Question

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Which of the following are trigonometric identities? Select all that apply. (3)
What is a simplified form of the expression sec^2 x-1/sinx secx?

Answer 1

Answer:

1. A, E

2. C

Step-by-step explanation:

1. tan theta = 1/cot theta

  sec theta = 1/ cos theta

2. Sec^2x-1 = tan^2x

   sinx sec x = tanx

   = tan^2x/tanx = tan x

Answer 2

The identities are (a) tan(θ) = 1/cot(θ), (c) 1 -  sin^2(θ) = cos^2(θ) and (e) sec(θ) = 1/cos(θ) and the simplified form of $\frac{\sec^2(x) - 1}{\sin(x)\sec(x)}$ is tan(x)

Part A: The trigonometry identities

As a general rule, we have:

tan(θ) = 1/cot(θ)

sec(θ) = 1/cos(θ)

cosec(θ) = 1/sin(θ)

sin^2(θ) + cos^2(θ) = 1

The above means that:

(a) tan(θ) = 1/cot(θ), (c) 1 -  sin^2(θ) = cos^2(θ) and (e) sec(θ) = 1/cos(θ) are identities

Part B: The trigonometric proof

We have:

$\frac{\sec^2(x) - 1}{\sin(x)\sec(x)}$

Express the numerator as tan^2(x)

$\frac{\tan^2(x)}{\sin(x)\sec(x)}$

The product sin(x)sec(x)  = tan(x).

So, we have:

$\frac{\tan^2(x)}{\tan(x)}$

Divide

tan(x)

Hence, the simplified form of $\frac{\sec^2(x) - 1}{\sin(x)\sec(x)}$ is tan(x)

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