In order to show that relative de Rham cohomology is a topological invariant, an equivalent of whitney's approximation theorem is needed for applications $f \colon (M,P) \to (N,Q)$ (i.e. $f(P) \subset Q$), where $M$, $N$ are smooth manifolds, $P \subset M$ a closed submanifold, and $Q \subset N$ a closed submanifold.


Do we have :


1) any continuous $f \colon (M,P) \to (N,Q)$ is (continuously) homotopic to a smooth $\tilde{f} \colon (M,P) \to (N,Q)$


2) if $f,\,g \colon (M,P) \to (N,Q)$ are smooth and (continuously) homotopic, they are smoothly homotopic.


I think it is true, but can't find any reference.


Same question for the proper version of the theorem. Do we have :


1) A proper continuous $f \colon (M,P) \to (N,Q)$ is (continuously) properly homotopic to a smooth proper $\tilde{f} \colon (M,P) \to (N,Q)$


2) if $f,\,g \colon (M,P) \to (N,Q)$ are smooth proper and (continuously) properly homotopic, they are smoothly and properly homotopic.



Is there a Whitney's approximation theorem for applications of pairs?
A good source is: 


* MR0674117 Reviewed Bröcker, Theodor; Jänich, Klaus Introduction to differential topology. Translated from the German by C. B. Thomas and M. J. Thomas. Cambridge University Press, Cambridge-New York, 1982. vii+160 pp.


They do this in detail. The relative situation is only dealt with for manifolds with boundary there, but the proofs can easily be adapted:
Choose a Riemannian metric on $N$ such that $Q$ is a totally geodesic submanifold and use the exponential mapping of this metric 
in the relevant proofs. 


