Verify the identity. Justify each step.tan θ+cot θ=1sin θ cos θ show all work in short terms

Answer 1

We start with the given expression:

tan θ + cot θ

Recall that tan θ = sin θ / cos θ and cot θ = cos θ / sin θ:

= sin θ / cos θ + cos θ / sin θ

Now, find a common denominator by multiplying the first term by sin θ / sin θ and the second term by cos θ / cos θ:

= sin^2 θ / (cos θ sin θ) + cos^2 θ / (cos θ sin θ)

Combine the fractions:

= (sin^2 θ + cos^2 θ) / (cos θ sin θ)

Recall the Pythagorean identity sin^2 θ + cos^2 θ = 1:

= 1 / (cos θ sin θ)

= 1 / sin θ cos θ

= 1/(sinθ * cosθ)

Therefore, tan θ + cot θ equals 1/sinθ*cosθ or 1.