The value of (sin⁴ 45°+ cos⁴ 60°) + (tan⁴ 45°+ cot⁴ 45°) is:

Given:

(sin4 45°+ cos4 60°) + (tan4 45°+ cot4 45°) 

Concept Used:

sin 45° = \(\frac{1}{\sqrt{2}}\), cos 60° = \(\frac{1}{2}\), tan 45° = 1, cot 45° = 1 

Calculation:

⇒ According to the question,

⇒ (sin4 45°+ cos4 60°) + (tan4 45°+ cot4 45°) 

⇒ Putting the values of the angles in the above equation we get,

⇒ (\(\frac{1}{\sqrt{2}}\))⁴ + (\(\frac{1}{2}\))⁴ + (1 + 1) = \(\frac{1}{4}\) + \(\frac{1}{16}\) + 2

⇒ \(\frac{4+1+2\times16}{16}\) = \(\frac{4+1+32}{16}\) = \(\frac{37}{16}\)

Therefore, the value of (sin4 45°+ cos4 60°) + (tan4 45°+ cot4 45°) is \(\frac{37}{16}\).