Basic Trigonometric Identities

We need to simplify the given trigonometric expression. sec^2 θ - tan^2 θ We can do it using one of the Pythagorean Identities. Let's start by recalling it! 1+tan^2 θ = sec^2 θ With this identity, we can substitute 1+tan^2 θ for sec^2 θ in our expression and simplify it.

Therefore, the given expression simplifies to 1. sec ^2 θ - tan ^2 θ = 1

We are asked to simplify the following trigonometric expression. sin θ sec θ/cot θ We can do it by using the Reciprocal Identities and the Tangent Identity. Let's start by recalling two of the Reciprocal Identities. sec θ = 1/cos θ, cos θ ≠ 0 [1.1em] tan θ = 1/cot θ, cot θ ≠ 0 Since we have cotθ in our expression, let's rewrite the second identity to express cotθ in terms of tanθ. tan θ = 1/cot θ ⇕ cot θ = 1/tan θ Now, we can substitute 1cos θ for sec θ and 1tan θ for cot θ in our expression. Then we will simplify it.

Finally, let's use the Tangent Identity. tan θ = sin θ/cos θ, cos θ ≠ 0 With this identity, we can substitute tan θ for sin θcos θ in our expression and simplify it.

We have simplified the given expression. sin θ sec θ/cot θ = tan ^2 θ

We are asked to check if the given equation is a trigonometric identity. sinθtanθ=1/cosθ-cosθ Let's start on the left-hand side and use Trigonometric Identities to rewrite it. If we obtain the expression on the right-hand side, then the equation is an identity. Let's recall the Tangent Identity. tanθ = sin θ/cos θ With this identity, we can substitute sin θcos θ for tan θ into the expression on the left-hand side.

Now we will use one of the Pythagorean Identities. Note that we need to rearrange the terms so that we can use it for our expression. sin^2 θ + cos^2 θ = 1 ⇕ sin^2 θ = 1 - cos^2 θ Let's substitute 1 - cos^2 θ for sin^2 θ in our expression.

We started on the left-hand side of the equation and used Trigonometric Identities to arrive at the right-hand side. Therefore, the given equation is an identity.

We want to determine whether the given equation is a trigonometric identity. sec θ - cos θ = tan θ sin θ Similarly to Part A, we will start on the left-hand side and use Trigonometric Identities to try to arrive at the right-hand side. First, let's recall one of the Reciprocal Identities. sec θ = 1/cos θ With this identity, we can substitute 1cos θ for sec θ in the left-hand side.

Now we will use one of the Pythagorean Identities. Note that we need to rearrange the terms so that we can use it for our expression. sin^2 θ + cos^2 θ = 1 ⇕ sin^2 θ = 1 - cos^2 θ Let's substitute sin^2 θ for 1 - cos^2 θ in our expression.

Finally, let's apply the Tangent Identity. tan θ = sinθ/cos θ With this identity, we can substitute tan θ for sin θcosθ in our expression.

We started on the left-hand side of the equation and by using Trigonometric Identities obtained the expression on the right-hand side. Therefore, the given equation is always true and, by definition, is an identity. sec θ - cos θ = tan θ sin θ ✓

We can start by recalling the Pythagorean Identity that relates cotangent and cosecant. 1 + cot^2 θ = csc^2 θ Let's rewrite this identity until cotθ is isolated on the left-hand side.

We have expressed cotθ in terms of csc θ.

We will use some of the Trigonometric Identities to rewrite tangent in terms of cosine. First, let's recall the Tangent Identity. tan θ = sin θ/cos θ Now we will use one of the Pythagorean Identities. Note that we will need to rearrange the terms so that we can use it in the expression we have in the right-hand side of the above identity. sin^2 θ + cos^2 θ = 1 ⇕ sin θ =± sqrt(1 - cos^2 θ) Now we can substitute ± sqrt(1 - cos^2 θ) for sin θ in tan θ= sin θcos θ.

We have written tan θ in terms of cos θ. tan θ = ± sqrt(1 - cos^2 θ)/cos θ