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.
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
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