Which of the following will satisfy a² = b² + (ab)² for the values a and b?

Solution (By Examveda Team)

We need to find which option makes the equation a² = b² + (ab)² true.
Remember, we are dealing with trigonometric functions: sin(x), cos(x), tan(x), and cot(x).
Let's look at each option:
Option A: a = sin(x), b = cot(x)
This means our equation becomes: sin²(x) = cot²(x) + (sin(x) * cot(x))²
Remember that cot(x) = cos(x) / sin(x). So, we can rewrite the equation.
sin²(x) = (cos²(x) / sin²(x)) + (sin(x) * (cos(x) / sin(x)))²
sin²(x) = (cos²(x) / sin²(x)) + cos²(x)
This looks complicated, and it's unlikely to simplify easily to something that's always true.
Option B: a = cos(x), b = tan(x)
This means our equation becomes: cos²(x) = tan²(x) + (cos(x) * tan(x))²
Remember that tan(x) = sin(x) / cos(x). Substitute that in.
cos²(x) = (sin²(x) / cos²(x)) + (cos(x) * (sin(x) / cos(x)))²
cos²(x) = (sin²(x) / cos²(x)) + sin²(x)
Again, this doesn't look like it will easily simplify to a true statement.
Option C: a = cot(x), b = cos(x)
This means our equation becomes: cot²(x) = cos²(x) + (cot(x) * cos(x))²
Substitute cot(x) = cos(x) / sin(x).
(cos²(x) / sin²(x)) = cos²(x) + ((cos(x) / sin(x)) * cos(x))²
(cos²(x) / sin²(x)) = cos²(x) + (cos²(x) / sin²(x)) * cos²(x)
(cos²(x) / sin²(x)) = cos²(x) + (cos⁴(x) / sin²(x))
Let's try to manipulate this a bit. Multiply both sides by sin²(x):
cos²(x) = cos²(x)sin²(x) + cos⁴(x)
cos²(x) = cos²(x)(sin²(x) + cos²(x))
Remember the fundamental trigonometric identity: sin²(x) + cos²(x) = 1
cos²(x) = cos²(x) * 1
cos²(x) = cos²(x). This is always true!
Option D: a = sin(x), b = tan(x)
This means our equation becomes: sin²(x) = tan²(x) + (sin(x) * tan(x))²
Substitute tan(x) = sin(x) / cos(x).
sin²(x) = (sin²(x) / cos²(x)) + (sin(x) * (sin(x) / cos(x)))²
sin²(x) = (sin²(x) / cos²(x)) + (sin⁴(x) / cos²(x))
This also doesn't look like it will simplify easily.
Therefore, Option C is the correct answer. It's the only one that simplifies to a true identity.