61. cos(x + yx) =
(A) cos (x) cos(yx) — sin (x) sin (yx)
(C) cos (x) sin (yx) — sin(x) cos (yx)
(B) cos (x) cos(yx) + sin (x) sin (yx)
(D) cos (x) sin (yx) + sin (x) cos (yx)
cos(x + yx) Formula
Correct Answer
Option (A): cos(x) cos(yx) − sin(x) sin(yx)
Detailed Explanation
The given expression is of the form cos(A + B), where the first angle is A = x and the second angle is B = yx. The standard trigonometric identity for the cosine of the sum of two angles is universally given as:
cos(A + B) = cos(A) cos(B) − sin(A) sin(B)
Substituting the given values into this identity, we obtain:
cos(x + yx) = cos(x) cos(yx) − sin(x) sin(yx)
This expression exactly matches Option (A). Hence, Option (A) is the correct answer.
The cosine addition identity is derived from the geometry of the unit circle and coordinate transformations. It plays a significant role in solving problems involving wave motion, Fourier analysis, signal processing, physics, engineering mathematics, and higher calculus. Since this identity is fundamental, it is frequently used to simplify complicated trigonometric expressions and solve equations efficiently.
Explanation of Every Option
Option (A): cos(x) cos(yx) − sin(x) sin(yx)
This option is correct because it follows the standard angle addition identity for cosine exactly. Whenever the angle is expressed as a sum, the cosine formula always contains a negative sign between the two products.
Option (B): cos(x) cos(yx) + sin(x) sin(yx)
This option is incorrect because the positive sign belongs to the cosine subtraction identity rather than the addition identity. In fact, this expression represents:
cos(A − B) = cos(A) cos(B) + sin(A) sin(B)
Therefore, Option (B) corresponds to cosine of the difference of two angles, not their sum.
Option (C): cos(x) sin(yx) − sin(x) cos(yx)
This option is incorrect because it mixes sine and cosine terms in an arrangement that does not represent the cosine addition formula. Instead, expressions of this form are associated with sine difference identities.
Option (D): cos(x) sin(yx) + sin(x) cos(yx)
This option is also incorrect because it represents the standard identity for the sine of the sum of two angles:
sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
Hence, this option belongs to the sine addition formula and not the cosine addition formula.
Final Answer
cos(x + yx) = cos(x) cos(yx) − sin(x) sin(yx)
Correct Option: (A)
