In her article “Patriarchy and Play in The Taming of the Shrew,” Marianne L. Novy of the University of Pittsburgh points out the connection between the induction and the first scene of the play-within-a-play. According to Novy,
As the “real” lord entertains us by showing that Sly can take a completely different place in the social order, the play begins to raise the question of how much that social order is a human construction whose validity is more like that of the game than that of divine or natural law. In the first scene of the inner play, the easy role change between Lucentio and Tranio, a servant clever enough to hide his precise degree of initiative from his master, repeats that question.
Indeed, the play-within-a-play begins with at least two differences between displayed and actual identity. The exchange of clothing, and therefore of identities, by Lucentio and Tranio is precipitated by Lucentio’s experience of love at first sight as he observes Bianca, the younger sister of the shrewish Katherine. His infatuation with Bianca, however, arises out of a false first impression.
As he describes her after his first observation of her, “sacred and sweet was all I saw in her.” In the end, however, Bianca reveals herself as neither “sacred” nor “sweet.” While exchanging barbs with the other characters at the wedding banquet in the final scene, Bianca offers a series of jests with defiant overtones and bawdy undertones. In a reference to the common Elizabethan image of a horned cuckold, Bianca takes a jab at the other newly-married couples, Petruchio with her sister Kate and Hortensio, her former suitor, with the Widow:
Head, and butt! An hasty-witted body
Would say your head and butt were head and horn.
When Petruchio offers to bring her into the exchange of comical insults, Bianca responds with a double entendre, indicating her current intention to drop out of the conversation through reference to sexual imagery:
Am I your bird? I mean to shift my bush;
And then pursue me as you draw your bow.
At this, she exits the room along with the other women at the banquet.
Bianca completes the revelation of her true, and certainly less than “sacred and sweet,” identity in her refusal of obedience to her new husband. When called upon by Lucentio to depart from her female friends and join him at the banquet, she sends the message through the servant Biondello “that she is busy and she cannot come.” Having been forced to rejoin her husband by the newly-tamed Katherine, Bianca offers mockery to the very notion that she has a “duty” to obey her husband:
Fie, what a foolish duty call you this?
I would your duty were as foolish [as Katherine’s], too.
The wisdom of your duty, fair Bianca,
Hath cost me a hundred crowns since suppertime.
The more fool you, for laying on my duty.
In this public chiding of her husband for having relied upon her to fulfill her duties as his wife, Bianca reveals herself as neither “sacred” nor “sweet,” contrary to Lucentio’s initial impression of her.
Lucentio’s mistaken first impression of Bianca leads to the exchange of identities between Lucentio and Tranio. Like the metamorphosis of Christopher Sly, this change of identities involves a change in apparent social class through a change of clothing. Tranio, slyly masking his desire for upward mobility in advice for his master, proposes an exchange of identities that will allow Lucentio, pretending to be a schoolmaster, to be close to Bianca and will allow Tranio to live the life proper to the son of the wealthy merchant Vincentio. In the form of a series of questions designed to bait his master into agreeing to the exchange of identities as if it were his own idea, Tranio makes the offer to Lucentio to
bear your part
And be in Padua here Vincentio’s son,
Keep house and ply his book, welcome his friends,
Visit his countrymen, and banquet them?
The two immediately exchange clothing to make real the exchange of identities. Significantly, Tranio has tricked his master and assumed his duties and lifestyle. Just as Sly’s ability to fill the role of a lord raises questions about the permanence of the inherited social order, Tranio’s ability to mislead his young master and to fulfill his master’s duties implies an innate equality of nature that surpasses inequalities imposed by differences in wealth and social standing derived from birth.
Later, Tranio plans and carries out another trick which upsets the social order by convincing a passing Pedant to masquerade as the wealthy merchant Vincentio, Lucentio’s father. The Pedant, disguised as Vincentio, is, like Tranio, able to fulfill the duties requisite to his newly-assumed social station, passing himself off as Vincentio even to Baptista, the father of Katherine and Bianca and a wealthy merchant himself. The trick works so well that Baptista guarantees Bianca’s hand in marriage to Tranio disguised as Lucentio in his meeting with the Pedant disguised as Vincentio. Commenting that he is “please[d]” by the Pedant’s “plainness and . . . shortness,” Baptista proclaims that “the match is made, and all is done. / Your son shall have my daughter with consent.”
Like the truth about Bianca, the true identities of Tranio and the Pedant are revealed in the final act as Tranio’s scheme unravels due to the appearance of the real Vincentio in Padua. Upon his arrival, the real Vincentio encounters the false Vincentio as a crowd gathers to “stand aside and see the end of this controversy.” As Tranio enters the scene already in progress, Vincentio already having been denied by both the false Vincentio and Biondello, the first words addressed to Tranio by Vincentio reference his clothing and the social status they falsely claim for him:
What am I, sir? Nay, what are you, sir? O immortal gods! Oh, fine villain! A silken doublet, a velvet hose, a scarlet cloak, and a copintank hat!
Only Gremio is able to recognize the real Vincentio, though even he remains confused by the identity of Tranio as Lucentio. As a result of this denial of recognition by others, the real Vincentio briefly loses his social status and is nearly arrested. As the officer comes to take him to the jail for impersonating himself, Vincentio exclaims in frustration that “thus strangers may be haled and abused,” realizing his loss of recognition as a loss of social status, and therefore a loss of identity.
The situation is resolved, however, and the real identities of Vincentio, Tranio, Lucentio, and the Pedant at last publicly revealed when the real Lucentio, now married to Bianca through a secret wedding ceremony, arrives and confesses the scheme to all present. Tranio, humiliated and restored to his former low position on the social scale, appears again as a servant in the following and final scene. There, the former mastermind behind the grand scheme to woo Bianca for Lucentio speaks only twice and both times addresses Petruchio, once again his social superior, as “sir.” While the question of equality has been raised, the stability of the Elizabethan social structure has been affirmed.
 Marianne L. Novy, “Patriarchy and Play in The Taming of the Shrew,” in Bloom, Modern Critical Interpretations, 14.
 The Taming of the Shrew, 1.1.176.
 Ibid., 5.2.40-41.
 Ibid., 5.2.46-47.
 Ibid., 5.2.86.
 Ibid., 1.1.195-198.
 Ibid., 4.4.38, 46-47.
 Ibid., 5.1.57.
 Ibid., 5.1.60-62.
 Ibid., 5.1.101.
 Ibid., 5.2.52, 55.