If tan(x) = 1, what is the value of sin(x) + cos(x)?

If tan(x) = 1, what is the value of sin(x) + cos(x)?

Correct Answer: √2

Step 1: Understand that tan(x) = sin(x) / cos(x).
Step 2: If tan(x) = 1, then sin(x) must equal cos(x) because sin(x) / cos(x) = 1 means they are the same.
Step 3: Let sin(x) = cos(x) = k for some value k.
Step 4: Since sin^2(x) + cos^2(x) = 1, we can write k^2 + k^2 = 1.
Step 5: This simplifies to 2k^2 = 1.
Step 6: Divide both sides by 2 to get k^2 = 1/2.
Step 7: Take the square root of both sides to find k = 1/√2.
Step 8: Now, since sin(x) = k and cos(x) = k, we have sin(x) + cos(x) = k + k = 2k.
Step 9: Substitute k = 1/√2 into 2k to get 2(1/√2).
Step 10: Simplify 2(1/√2) to get √2.

Trigonometric Identities – Understanding the relationship between sine, cosine, and tangent functions.
Pythagorean Identity – Using the identity sin²(x) + cos²(x) = 1 to derive values.
Angle Values – Recognizing specific angle values where trigonometric functions yield known results.