Enter your answer and show all the steps that you use to solve this problem in the space provided.
Verify the identity. Justify each step.
tan θ+cot θ=1sin θ cos θ
We start with the given trigonometric identity:
tan θ + cot θ = 1 / (sin θ cos θ)
Now, we know that tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Substituting these values into the equation, we have:
(sin θ / cos θ) + (cos θ / sin θ) = 1 / (sin θ cos θ)
To simplify this expression, we first find a common denominator for the left side:
(sin^2 θ + cos^2 θ) / (sin θ cos θ) = 1 / (sin θ cos θ)
Using the Pythagorean identity sin^2 θ + cos^2 θ = 1, we have:
1 / (sin θ cos θ) = 1 / (sin θ cos θ)
Therefore, the given trigonometric identity is verified. Each step was justified by applying the definition of tan and cot, finding a common denominator, and using the Pythagorean identity.