If cos 37° = x, then find the value of tan 53°.

We know that:

• tan 53° = tan (90° - 37°)
• Using the trigonometric identity, tan (90° - θ) = cot θ:
• tan 53° = cot 37°

By the definition of cotangent: 
cot 37° = cos 37° / sin 37°

Substituting the value of cos 37° as x, we get: 
cot 37° = x / sin 37° ......(i)

Now, to find the value of sin 37°, we use the Pythagorean identity: 
sin² 37° + cos² 37° = 1

Substituting cos² 37° = x², we have: 
sin² 37° = 1 - x²

Taking the square root: 
sin 37° = √(1 - x²)

Substituting this value of sin 37° into equation (i), we get: 
cot 37° = x / √(1 - x²)

Hence: 
tan 53° = cot 37° = x / √(1 - x²)

Final Answer:

tan 53° = x / √(1 - x²)