The Society states regarding the translation of the words in question:
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Often, the different shades of meaning of pi·steu′o must be discerned from the context. At times, though, different grammatical constructions help us to see what the writer had in mind. For example, if pi·steu′o is followed merely by a noun in the dative case, the New World Translation usually renders it simply as “believe”—unless the context indicates something different. (Matthew 21:25, 32; but see Romans 4:3.) If pi·steu′o is followed by the word e·pi′, “on,” it is generally rendered “believe on.” (Matthew 27:42; Acts 16:31) If it is followed by eis, “to,” it is usually translated “exercise faith in.”—John 12:36; 14:1.
This latter rendering (which reminds us that pi·steu′o is related to the Greek word pi′stis, “faith”) is in harmony with a comment in An Introductory Grammar of New Testament Greek, by Paul Kaufman. This work says: “Another construction which is common in the New Testament (especially in John’s Gospel) is πιστευω [pi·steu′o] with εις [eis] and the accusative case . . . The whole construction of εις plus the accusative must be translated rather than attempting to translate the preposition εις as an isolated word. Faith is thought of as an activity, as something men do, i.e. putting faith into someone.”
But why "exercise faith in" instead of "trust in" or "place trust in"? The latter gets at the subtle nuance and construes belief as an activity. The former is an overtranslation that adds further nuances. Exercise implies effort. One must exert oneself. And that exertion would involve other actions one would do in order to exercise faith. And those acts must continually be performed. This facilitates an interpretation that replaces faith with works (specifically, the preaching work):
*** km 5/70 p. 1 par. 2 Being Well-Pleasing to God *** Accordingly, to have Jehovah’s favor and approval on ourselves and our families, and to expect our ministerial work to prosper, we must exercise faith—faith in the leadership of our Lord Jesus Christ, whom Jehovah sent forth. This faith will impel us to do the same work that Jesus did. He had faith in his Father, knowing that God was with him. Jehovah and Jesus Christ are more interested than we are in ourselves and in the Kingdom interests we are caring for. Consequently, we can be sure that God will prosper our work that is done in faith.
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Thus, once you are in the way of salvation, exert yourself to stay in it. You cannot earn salvation by works; faith is what is needed. (Rom. 3:10-12; Gal. 3:11) But you do need to exercise your faith by practical works that give proof of it. (Jas. 2:14-17, 26) That is what the apostles did. They especially gave emphasis to the preaching work that Jesus commanded be done by Christians.
*** w76 12/15 p. 760 pars. 22-23 Gain the Promised Eternal Life—Exercise Faith! ***
In order to enter into the fulfillment of God's promise we must possess life; we need to have our souls preserved alive. We can gain that prize of life solely by a sustained faith. We eagerly desire to enjoy God’s fulfilled promise eternally. Away, then, with any thought or inclination to shrink back in fear and in unbelief! Faith is what we will exercise along with works in proof! In reward for that, Jehovah God the Life-Giver will preserve our souls alive forever.—1 John 2:25.
Without fail, therefore, the "God who supplies endurance" will fulfill his promise to the faithfully enduring doers of his will. (Rom. 15:5) Joyfully he will usher us into the eternal blessings and privileges of his long-promised kingdom by his Son Jesus Christ. Thus, not in vain shall we have preached this kingdom "in all the inhabited earth for a witness to all the nations" before "the end" came.—Matt. 24:14.
*** w83 10/1 p. 14 par. 3 Millions Get Ready for Uninterrupted Life on Earth ***
To people now living on earth, the reigning King Jesus Christ becomes "the life" due to the fact that they exercise faith in him, this faith being proved by works. During this "conclusion of the system of things," these works would include taking an active part in the fulfillment of Jesus' prophecy at Matthew 24:14 to preach the good news of God’s Kingdom before "the end" of the system comes. (Matthew 24:3; Mark 13:3, 4) The exercisers of faith who survive the end of the current system of things and who look forward to life in Paradise restored will never need to die at all. (1 John 2:17)
*** w05 5/1 p. 29 Questions From Readers ***
Many scriptures show that to be saved, a person must come to know Jehovah, believe in Jesus, and exercise faith, demonstrating that faith by works.
All of this goes very far from the sense of trusting in someone. The overtranslation of "believe/trust in" as "exercise faith in" also combines with the overtranslation of "knowing" as "putting in knowledge" and "declaring" as "making public declaration". All of that is interpreted as pertaining to the preaching work. The overtranslations exist because they facilitate a certain desired soteriology, which goes beyond having a personal dedication and commitment to Jesus.