secθ - secθsin^2θ = _____0sin^2θsinθtanθcosθ

To solve the given equation, we will simplify both sides step by step.

Starting with the left side of the equation: secθ - secθsin^2θ

We can factor out secθ from both terms: secθ(1 - sin^2θ)

Using the Pythagorean identity sin^2θ + cos^2θ = 1, we can rewrite the expression as: secθ(cos^2θ)

Now, let's simplify the right side of the equation: 0sin^2θsinθtanθcosθ

Since 0 multiplied by any term is always 0, we can simplify this expression to 0.

Therefore, the equation becomes: secθ(cos^2θ) = 0

Now, we have two possibilities for the equation to be true:

1. secθ = 0
2. cos^2θ = 0

For the first possibility, secθ = 0, we know that secθ is equal to 1/cosθ. So, 1/cosθ = 0, which is not possible since division by zero is undefined. Therefore, secθ cannot be equal to 0.

For the second possibility, cos^2θ = 0, we can take the square root of both sides: cosθ = 0

This means that θ is equal to π/2 + nπ or 3π/2 + nπ, where n is an integer.

In conclusion, the equation secθ - secθsin^2θ = 0sin^2θsinθtanθcosθ is satisfied when θ is equal to π/2 + nπ or 3π/2 + nπ, where n is an integer.