Explanation Given: tan x + tan y 1 − tan x tan y = sin x cos y + cos x sin y cos x cos y − sin x sin y Formula used; tan x = sin x cos x Proof: Consider the left side of the given expression: tan x + tan y 1 − tan x tan y Using the quotient identity of trigonometry tan x = sin x cos x and tan y = sin y cos y , the above expression can be further simplified as: tan x + tan y 1 − tan x tan y = sin x cos x + sin y cos y 1 − sin x cos x . sin y cos y Express the terms in the numerator and denominator with least common denominator cos x cos y : sin x cos x + sin y cos y 1 − sin x cos x

5th Edition

ISBN: 9780321837240

Author: Robert F. Blitzer

Publisher: PEARSON

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Chapter 6.1, Problem 43E

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To prove: The identity tanx+tany1−tanxtany=sinxcosy+cosxsinycosxcosy−sinxsiny.

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Chapter 6.1, Problem 42E

Chapter 6.1, Problem 44E

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R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]

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Chapter 6 Solutions

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Ch. 6.1 - Fill in each blank so that the resulting statement...Ch. 6.1 - Fill in each blank so that the resulting statement...Ch. 6.1 - Fill in each blank so that the resulting statement...Ch. 6.1 - Fill in each blank so that the resulting statement...Ch. 6.1 - Fill in each blank so that the resulting statement...Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E

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To prove: The identity tan x + tan y 1 − tan x tan y = sin x cos y + cos x sin y cos x cos y − sin x sin y .

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To prove: The identity tanx+tany1−tanxtany=sinxcosy+cosxsinycosxcosy−sinxsiny.

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Chapter 6 Solutions