Your main idea is more or less correct. The cosine hidden in the projected solid angle measure $$dA^\perp = dA\cos(θ)$$ compensates the weakening of irradiance due to inciden angle (the Lambert's cosine law). This makes radiance independent from the incident angle. Mi guess is that the main motivation was to make it more practical to work with.

The cosine in the rendering equation comes from the definition of BRDF:

$$ f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{\mathrm{d}E\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\cos\theta_{i}\mathrm{d}\omega_{i}} $$

Which can be rewritten as

$$ \mathrm{d}L_{o}\left(\omega_{o}\right) =f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Which can be integrated over the hemisphere the get the radiance reflected in a given direction $\omega_o$

$$ \int_\Omega f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Voila, the rendering equation!

PS: Sorry about the changed notation ($\omega_{i}, \omega_{o}$ vs. $\omega^{\prime}, \omega$). I find this one more practical. I can rewrite it if it confuses you.

Your main idea is more or less correct. The cosine hidden in the projected area measure $dA^\perp = dA\cos(θ)$ compensates the weakening of irradiance due to incident angle (the Lambert's cosine law). This makes radiance independent from the incident angle. My guess is that the main motivation was to make it more practical to work with.

The cosine in the rendering equation comes from the definition of BRDF:

$$ f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{\mathrm{d}E\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\cos\theta_{i}\mathrm{d}\omega_{i}} $$

Which can be rewritten as

$$ \mathrm{d}L_{o}\left(\omega_{o}\right) =f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Which can be integrated over the hemisphere the get the radiance reflected in a given direction $\omega_o$

$$ \int_\Omega f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Voila, the rendering equation!

Conclusion: Althought both cosines come from the same fundamental priciple, they serve different purposes.

Your main idea is more or less correct. The cosine hidden in the projected solid angle measure $$dA^\perp = dA\cos(θ)$$ compensates the weakening of irradiance due to inciden angle (the Lambert's cosine law). This makes radiance independent from the incident angle. Mi guess is that the main motivation was to make it more practical to work with.

The cosine in the rendering equation comes from the definition of BRDF:

$$ f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{\mathrm{d}E\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\cos\theta_{i}\mathrm{d}\omega_{i}} $$

Which can be rewritten as

$$ \mathrm{d}L_{o}\left(\omega_{o}\right) =L_{i}\left(\omega_{i}\right) f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Which can be integrated over the hemisphere the get the reflected radiance in a given direction $\omega_o$

$$ \int_\Omega L_{i}\left(\omega_{i}\right) f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Voila, the rendering equation!

PS: Sorry about the changed notation. I find this one more practical. I can rewrite it if it confuses you.

Your main idea is more or less correct. The cosine hidden in the projected solid angle measure $$dA^\perp = dA\cos(θ)$$ compensates the weakening of irradiance due to inciden angle (the Lambert's cosine law). This makes radiance independent from the incident angle. Mi guess is that the main motivation was to make it more practical to work with.

The cosine in the rendering equation comes from the definition of BRDF:

$$ f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{\mathrm{d}E\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)} =\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\cos\theta_{i}\mathrm{d}\omega_{i}} $$

Which can be rewritten as

$$ \mathrm{d}L_{o}\left(\omega_{o}\right) =f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Which can be integrated over the hemisphere the get the radiance reflected in a given direction $\omega_o$

$$ \int_\Omega f_{r}\left(\omega_{i}\rightarrow\omega_{o}\right) L_{i}\left(\omega_{i}\right) \cos\theta_{i} \mathrm{d}\omega_{i} $$

Voila, the rendering equation!

PS: Sorry about the changed notation ($\omega_{i}, \omega_{o}$ vs. $\omega^{\prime}, \omega$). I find this one more practical. I can rewrite it if it confuses you.