`\color{blue} ✍️` In this Section, we shall derive expressions for trigonometric functions of the sum and difference of two numbers (angles) and related expressions.
The basic results in this connection are called trigonometric identities. We have seen that
`1. \ \ color{red}(sin (– x) = – sin x)`
`2. \ \ color{red}( cos (– x) = cos x)`
We shall now prove some more results:
`3. \ \ color{red}( cos (x + y) = cos x cos y – sin x sin y)`
`\color{blue} ✍️` Consider the unit circle with centre at the origin. Let `x` be the angle `color{red}(P_4OP_1)` and `y` be the angle `color{red}(P_1OP_2)`. Then `color{red}((x + y))` is the angle `color{red}(P_4OP_2)`. Also let `color{red}((– y))` be the angle `color{red}(P_4OP_3)`.
`\color{blue} ✍️` Therefore, `P_1, P_2, P_3` and `P_4` will have the coordinates `color{red}(P_1 \ \(cos x, sin x), ``color{greeen}[ P_2 \ \ [cos (x + y), sin (x + y)]]`,`color{purple}[ P_3 \ \ [cos (– y), sin (– y)]]` and `color{red}(P_4 \ \ (1, 0))` (Fig 3.14).
Consider the triangles `color{red}(P_1OP_3)` and `color{red}(P_2OP_4)`. They are congruent Therefore,
`color{red}(1P_1P_3)` and `color{red}(1P_2P_4)` are equal. By using distance formula, we get
`color{red}(1P_1P_3^2 = [cos x – cos (– y)]^2 + [sin x – sin(–y)^2]`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= (cos x – cos y)^2 + (sin x + sin y)^2`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= cos^2 + cos^2 y – 2 cos x cos y + sin^2 x + sin^2 y + 2sin x sin y`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= 2 – 2 (cos x cos y – sin x sin y)`
Also, `color{red}(P_2P_4^2 = [1 – cos (x + y)]^2 + [0 – sin (x + y)]^2)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= 1 – 2cos (x + y) + cos^2 (x + y) + sin^2 (x + y)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= 2 – 2 cos (x + y)`
Since `color{red}(P_1P_3 = P_2P_4,)` we have `color{red}(P_1P_3^2 = P_2P_4^2)`.
Therefore, `2 –2 (cos x cos y – sin x sin y) = 2 – 2 cos (x + y)`.
Hence `color{red}(cos (x + y) = cos x cos y – sin x sin y)`
`\color{blue} ✍️` In this Section, we shall derive expressions for trigonometric functions of the sum and difference of two numbers (angles) and related expressions.
The basic results in this connection are called trigonometric identities. We have seen that
`1. \ \ color{red}(sin (– x) = – sin x)`
`2. \ \ color{red}( cos (– x) = cos x)`
We shall now prove some more results:
`3. \ \ color{red}( cos (x + y) = cos x cos y – sin x sin y)`
`\color{blue} ✍️` Consider the unit circle with centre at the origin. Let `x` be the angle `color{red}(P_4OP_1)` and `y` be the angle `color{red}(P_1OP_2)`. Then `color{red}((x + y))` is the angle `color{red}(P_4OP_2)`. Also let `color{red}((– y))` be the angle `color{red}(P_4OP_3)`.
`\color{blue} ✍️` Therefore, `P_1, P_2, P_3` and `P_4` will have the coordinates `color{red}(P_1 \ \(cos x, sin x), ``color{greeen}[ P_2 \ \ [cos (x + y), sin (x + y)]]`,`color{purple}[ P_3 \ \ [cos (– y), sin (– y)]]` and `color{red}(P_4 \ \ (1, 0))` (Fig 3.14).
Consider the triangles `color{red}(P_1OP_3)` and `color{red}(P_2OP_4)`. They are congruent Therefore,
`color{red}(1P_1P_3)` and `color{red}(1P_2P_4)` are equal. By using distance formula, we get
`color{red}(1P_1P_3^2 = [cos x – cos (– y)]^2 + [sin x – sin(–y)^2]`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= (cos x – cos y)^2 + (sin x + sin y)^2`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= cos^2 + cos^2 y – 2 cos x cos y + sin^2 x + sin^2 y + 2sin x sin y`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= 2 – 2 (cos x cos y – sin x sin y)`
Also, `color{red}(P_2P_4^2 = [1 – cos (x + y)]^2 + [0 – sin (x + y)]^2)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= 1 – 2cos (x + y) + cos^2 (x + y) + sin^2 (x + y)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= 2 – 2 cos (x + y)`
Since `color{red}(P_1P_3 = P_2P_4,)` we have `color{red}(P_1P_3^2 = P_2P_4^2)`.
Therefore, `2 –2 (cos x cos y – sin x sin y) = 2 – 2 cos (x + y)`.
Hence `color{red}(cos (x + y) = cos x cos y – sin x sin y)`