Rewrite the expression tan(90°−θ)sinθ as one of the six trigonometric functions of acute angle thetaθ.

Please show work + answer. I have a few of them to do. Thanks.

tan(90º - Θ) = cotan(Θ)

cotan(Θ) = cos(Θ) / sin(Θ)

[cos(Θ) / sin(Θ)] * sin(Θ) = cos(Θ)

To rewrite the expression `tan(90°−θ)sinθ` as one of the six trigonometric functions, we can use trigonometric identities.

First, let's rewrite `tan(90°−θ)` using the identity `tan(90°−θ) = cotθ`:

`tan(90°−θ)sinθ = cotθsinθ`

Next, let's simplify `cotθsinθ` using the identity `cotθ = cosθ/sinθ`:

`cotθsinθ = (cosθ/sinθ)sinθ`

The `sinθ` terms cancel out:

`= cosθ`

Therefore, `tan(90°−θ)sinθ` is equivalent to `cosθ`.

To rewrite the expression tan(90° - θ)sinθ as one of the six trigonometric functions, we can use trigonometric identities.

First, let's recall two important trigonometric identities:

1. The cofunction identity: sin(90° - θ) = cosθ
2. The quotient identity: tanθ = sinθ / cosθ

Using these identities, we can rewrite the expression:

tan(90° - θ)sinθ

= (sin(90° - θ) / cos(90° - θ)) * sinθ

Now, substitute the cofunction identity sin(90° - θ) = cosθ:

= (cosθ / cos(90° - θ)) * sinθ

Next, recall another trigonometric identity: sinθ = 1/cscθ:

= (cosθ / cos(90° - θ)) * (1/cscθ)

Applying the cofunction identity cos(90° - θ) = sinθ:

= (cosθ / sinθ) * (1/cscθ)

Using the reciprocal identity cscθ = 1/sinθ:

= cosθ * (1/sinθ) * (1/cscθ)

This simplifies to:

= cosθ / sinθ

Finally, using the reciprocal identity for sine, sinθ = 1/cscθ:

= cosθ * cscθ

Therefore, the expression tan(90° - θ)sinθ is equivalent to cosθ * cscθ.