Answer :
We can prove the identity cot⁴a + cot²a = 1 by using the trigonometric identity that cot²a = cos⁻²a and then substituting cos⁴a + cos²a = 1 from the given equation.
Here's the derivation:
* Start with the identity: cot²a = cos⁻²a
* Substitute cos⁴a + cos²a = 1 (given) into the equation: cot²a = (cos²a)^-1 = (cos⁴a + cos²a)^-1
* Expand the denominator using the distributive property: cot²a = (cos⁴a + cos²a)^-1 = (cos⁴a)^-1 + (cos²a)^-1
* Simplify using the exponent rule (a^n)^m = a^(n*m): cot²a = cos⁻⁴a + cos⁻²a
Finally, since cot⁴a = cos⁻⁴a and cot²a = cos⁻²a, we have proven the identity cot⁴a + cot²a = 1.
Here's the derivation:
* Start with the identity: cot²a = cos⁻²a
* Substitute cos⁴a + cos²a = 1 (given) into the equation: cot²a = (cos²a)^-1 = (cos⁴a + cos²a)^-1
* Expand the denominator using the distributive property: cot²a = (cos⁴a + cos²a)^-1 = (cos⁴a)^-1 + (cos²a)^-1
* Simplify using the exponent rule (a^n)^m = a^(n*m): cot²a = cos⁻⁴a + cos⁻²a
Finally, since cot⁴a = cos⁻⁴a and cot²a = cos⁻²a, we have proven the identity cot⁴a + cot²a = 1.