To be responsible to truth is to encounter it/respond to it for what it is. Hamlet first manifested it in his adamant refusal to shed the outward trappings of mourning against the protestation of his uncle king and queen mother, who wanted him to express the jubilation of their new marriage. Stubborn as he is in his fidelity and sense of responsibility towards his deceased king father, he insisted in the truth of "that within which passeth show" (Shakespeare, Act 1, Scene II). As for Phaedra, when pressed by Oeneoneas to the cause of her depression, she made a simple acknowledgement of the truth raging within her: "I feel all the furies of desire". (Racine, Act 1 Scene III). No matter how fearsome the truth is, she shrinks not in acknowledging and recognizing its potent poison. Both of them yielded to the power of this truth but differ in the manner of their response. Hamlet refused to hide the truth; Phaedra chose to languish in secret till she could bear no more to hide the truth.
To be responsible to truth is to act so that that the truth comes to light, no matter what the price. When the deceased king's apparition revealed to Hamlet the dastardly act of murder committed by his uncle against his father and commanded him to execute vengeance, Hamlet subsequently manifested a prolonged ambivalence towards acting on the revelation. Far from fear of the challenge, Hamlet's tarrying was rather a strong proof of his commitment and responsibility for truth. His words: "prompted to my revenge by heaven and hell," (Shakespeare, Act III, Scene I) belied his confidence in the apparition's revelation and could not commit himself to such unverified disclosure. He, however, was committed to ascertain the truthfulness of the apparition's revelation, to the extent of feigning madness and suffering the negative consequences of such demeaning stance, if only to achieve his end.
Phaedra on the other hand, needs to discover the truth about the consequences of the truth she nurtures within her. She has from the start sensed the risks and dangers of giving in to this enticement towards the truth burning inside her. She realized with tremendous force the power of this truth to destroy - though it ravished her soul - that she exclaimed, "I recognized Venus and her fearsome fires" (Shakespeare, Act I, Scene III0. Her insight to the darkness that hangs upon her moved her to solicit with extraordinary entreaties the favor of the same Goddess whom she believed caused her torment, but found no relief from her. She tried all means to discover the truth and attempted " to banish the enemy" (Racine, Act 1 Scene 3) by urging the exile of Hippolytus; but upon his return she discovered the enemy not vanquished. Thinking that by yielding to this power within will conquer the enemy, she mustered whatever strength of will remained inside to unveil her dark secret in progressive step, first to her nurse (Racine, Act I, Scene III) and confidante and by the latter's help to the object of her desire - Hippolytus himself (Racine, Act II, Scene V). Only then did she discover a far heart wrenching truth that her truth does not match with the truth inside Hippolytus. Rather, the truth inside her became the subversion of all she hoped her truth