If cosA + sinA= root2 cosA
Show that cosA - sinA= root2 sinA
cosA + sinA= √2*cosA
=> sinA = √2*cosA - cosA
=> sinA = (√2 - 1)cosA
cosA - sinA = cosA - (√2 - 1)cosA
= √2*[cosA(√2 - 1)]
= √2*sinA
cosA + sinA= √2*cosA
(cosA + sinA)^2 = 2cos^2A
cos^2A + 2sinAcosA + sin^2A = 2cos^2A
cos^2A-sin^2A = 2sinAcosA
(cosA+sinA)(cosA-sinA) = 2sinAcosA
(√2*cosA)(cosA-sinA) = 2sinAcosA
cosA-sinA = √2*sinA
To show that cosA - sinA = √2sinA, let's start with the given equation:
cosA + sinA = √2cosA
Now, let's subtract sinA from both sides of the equation:
cosA + sinA - sinA = √2cosA - sinA
cosA = √2cosA - sinA
Next, let's isolate the √2cosA term by subtracting cosA from both sides of the equation:
cosA - cosA = √2cosA - cosA - sinA
0 = (√2cosA - cosA) - sinA
Now, let's factor out cosA from the first term on the right side:
0 = (cosA(√2 - 1)) - sinA
Finally, let's divide both sides of the equation by (√2 - 1) to solve for sinA:
0 = -sinA
Since we have 0 = -sinA, we can conclude that sinA = 0.
Now, let's substitute sinA = 0 back into the original equation to find the value of cosA:
cosA + 0 = √2cosA
cosA = √2cosA
Divide both sides of the equation by cosA:
1 = √2
Since this is not a valid equation, it means that sinA cannot be equal to 0.
Therefore, cosA - sinA ≠ √2sinA.
To show that cosA - sinA = root2 sinA, we can start by manipulating the given equation: cosA + sinA = root2 cosA.
1. Start with the given equation:
cosA + sinA = root2 cosA.
2. Subtract cosA from both sides:
sinA = root2 cosA - cosA.
3. To combine the terms on the right side, we need to find a common denominator. Since cosA is equivalent to (1/cosA) multiplied by cosA, we can rewrite the equation as follows:
sinA = [root2 cosA - (cosA/cosA)].
Simplify the equation:
sinA = [root2 cosA - 1].
4. Next, we want to transform sinA to cosA. We can use the Pythagorean identity: sin^2(A) + cos^2(A) = 1.
Rearrange the equation to solve for sinA:
sin^2(A) = 1 - cos^2(A).
sinA = sqrt(1 - cos^2(A)).
5. Substitute the value of sinA from step 3 into the equation obtained in step 4:
sqrt(1 - cos^2(A)) = [root2 cosA - 1].
6. Square both sides of the equation to eliminate the square root:
1 - cos^2(A) = [root2 cosA - 1]^2.
7. Expand and simplify the right side of the equation:
1 - cos^2(A) = (root2 cosA - 1) * (root2 cosA - 1).
1 - cos^2(A) = 2 cos^2(A) - 2 root2 cosA + 1.
8. Rearrange the equation and combine like terms:
cos^2(A) - 2 cos^2(A) + 2 root2 cosA - 1 = 0.
-cos^2(A) + 2 root2 cosA - 1 = 0.
9. Multiply all terms by -1 to change the sign of the equation:
cos^2(A) - 2 root2 cosA + 1 = 0.
10. Notice that this quadratic equation is similar to (cosA - root2 sinA)^2 = 0.
By comparing these two equations, we can see that cosA - root2 sinA = 0.
11. Rearrange the equation by adding root2 sinA to both sides to get:
cosA = root2 sinA.
12. Divide both sides of the equation by cosA to solve for sinA:
sinA = (root2 sinA)/cosA.
13. Simplify the equation to get:
sinA = root2 sinA.
Since sinA = root2 sinA, we have shown that cosA + sinA = root2 cosA implies that cosA - sinA = root2 sinA.