$\sin (90°-\theta )\sec (180°-\theta )\sin (180°+\theta )\tan (270°-\theta )\cot (90°+\theta )=$

The given trigonometric expression is:

sin(90° − θ) × sec(180° − θ) × sin(180° + θ) × tan(270° − θ) × cot(90° + θ)

Step 1: Use of Trigonometric Identities

First, we can simplify each trigonometric function using known identities:

• sin(90° − θ) is equivalent to cos θ because sin(90° − θ) = cos θ.
• sec(180° − θ) is equivalent to −sec θ because sec(180° − θ) = −sec θ.
• sin(180° + θ) is equivalent to −sin θ because sin(180° + θ) = −sin θ.
• tan(270° − θ) is equivalent to −cot θ because tan(270° − θ) = −cot θ.
• cot(90° + θ) is equivalent to −tan θ because cot(90° + θ) = −tan θ.

Step 2: Substituting Values

Substitute the simplified values into the original expression:

cos θ × (−sec θ) × (−sin θ) × (−cot θ) × (−tan θ)

Step 3: Simplifying the Expression

Now, let’s simplify the expression step by step:

• cos θ × −sec θ simplifies to −1 because cos θ × sec θ = 1 (sec θ is the reciprocal of cos θ).
• −sin θ × cot θ simplifies to −cos θ because cot θ = cos θ / sin θ.
• −cos θ × tan θ simplifies to −sin θ because tan θ = sin θ / cos θ.

Step 4: Final Simplification

Now, the expression looks like: −sin θ

Conclusion

Therefore, the simplified result of the given trigonometric expression is: −sin θ

Important Notes:

Throughout this solution, we have used the fact that:

• cos 90° = 0 and this is important in simplifying trigonometric functions involving multiples of 90°. Understanding the behavior of trigonometric functions for standard angles like 90° helps to simplify many expressions.
• Also, understanding the signs of trigonometric functions in different quadrants helps in solving such expressions effectively.

By using these identities, we were able to reduce the given expression to −sin θ.