Simplify the right-hand side of the equation using one of the Pythagorean Identities. Then simplify the left-hand side of the equation using the Quotient and the Reciprocal Identities to arrive at the right-hand side.
See solution.
Practice makes perfect
We want to verify the given trigonometric identity.
tan ^2 θ csc ^2 θ = 1 + tan ^2 θ
We will simplify the right-hand side of the equation using one of the Pythagorean Identities. Then we will simplify the left-hand side of the equation using the Quotient and the Reciprocal Identities to arrive at the right-hand side. Let's start by recalling one of the Pythagorean Identities.
tan ^2 θ +1 =sec^2 θWe can substitute sec^2 θ for 1+tan ^2 θ in our expression on the right-hand side.
Now we can move on to simplifying the expression on the left-hand side. Let's recall one of the Quotient Identities.
tan θ = sin θ/cos θ, cos θ ≠0
We can substitute sin θcos θ for tan θ in our expression.
Next, let's recall one of the Reciprocal Identities.
csc θ = 1/sin θ, sin θ ≠0
We will substitute 1sin θ for csc θ in our expression. Then we will continue simplifying.
Finally, let's recall once again one of the Reciprocal Identities.
sec θ = 1/cos θ, cos θ ≠0
We can substitute sec θ for 1cos θ in our expression.
We obtained the expression on the left-hand side that is equal to simplified form of the expression on the right-hand side. Therefore, we have verified the identity.
