What is the simplified value of the trigonometric expression sin^2(90° − θ) − [sin(90° − θ) · sin θ / tan θ]?

Introduction / Context:
This trigonometric simplification problem checks familiarity with complementary angle identities and the relationship between sine, cosine, and tangent. The expression may look complicated at first, but it reduces quickly if we systematically apply basic identities for 90° − θ and for tan θ.

Given Data / Assumptions:

Expression: sin^2(90° − θ) − [sin(90° − θ) · sin θ / tan θ].
θ is such that tan θ is defined and non zero (so sin θ and cos θ are not zero simultaneously and cos θ is non zero).

Concept / Approach:
We use the complementary angle identities: sin(90° − θ) = cos θ and cos(90° − θ) = sin θ. We also use tan θ = sin θ / cos θ. By replacing sin(90° − θ) with cos θ, the expression becomes easier to handle. Then we simplify the fraction involving sin θ divided by tan θ and look for cancellation.

Step-by-Step Solution:
First, replace sin(90° − θ) with cos θ.Then sin^2(90° − θ) becomes cos^2 θ.The expression becomes cos^2 θ − [cos θ · sin θ / tan θ].Recall that tan θ = sin θ / cos θ.Thus sin θ / tan θ = sin θ / (sin θ / cos θ) = cos θ, provided sin θ is not zero.So the bracketed term becomes cos θ · cos θ = cos^2 θ.Therefore the whole expression simplifies to cos^2 θ − cos^2 θ.Finally, cos^2 θ − cos^2 θ = 0.

Verification / Alternative check:
Choose a convenient angle, for example θ = 30°. Then sin(90° − 30°) = sin 60° = √3/2 and cos 30° = √3/2. Compute the original expression numerically: sin^2 60° = (√3/2)^2 = 3/4. The second term is sin 60° · sin 30° / tan 30°. Now sin 30° = 1/2, tan 30° = 1/√3. So sin 60° · sin 30° / tan 30° = (√3/2) * (1/2) / (1/√3) = (√3/4) * √3 = 3/4. Hence the expression is 3/4 − 3/4 = 0, confirming the algebraic simplification.

Why Other Options Are Wrong:
Values such as 1, cosec θ, cos θ, or sin θ would remain dependent on θ, while our simplification shows the expression collapses to a constant 0 for all admissible θ. Therefore any option that depends on θ cannot be correct, leaving only 0 as the consistent result.

Common Pitfalls:
A frequent error is to forget that sin(90° − θ) equals cos θ and instead confuse it with sin θ or tan θ. Another mistake is mishandling the division by tan θ, such as treating sin θ / tan θ as sin θ * tan θ rather than sin θ / (sin θ / cos θ). Keeping the identities clear and writing intermediate steps avoids these issues.

Final Answer:
The simplified value of the expression is 0.