To solve the given expression 1 + tan²(a) + 1 + cot²(a) + 1 + tan(a) + 1 + cot²(a) tan²(a), we’ll first rewrite it in a clearer format:
- Let tan²(a) = x, so cot²(a) = 1/tan²(a) = 1/x.
Now, substituting these in:
- The expression becomes: 4 + x + 1/x + tan(a) + 1/x * x
Notice that 1/x * x simplifies to 1. Therefore, our expression now reads:
- 4 + x + 1/x + tan(a) + 1
This simplifies to:
- 5 + x + 1/x + tan(a)
Next step is to manipulate the terms:
- We know from trigonometric identities that tan(a) = sin(a)/cos(a).
- Therefore, tan²(a) + cot²(a) = 2 + 2 tan²(a)cot²(a) following the Pythagorean identity.
Thus, we switch gears a bit: evaluate the function 5 + tan²(a) + cot²(a), where both are based upon the angle ‘a’ .
Using the identity, where:
- 1 + tan²(a) = sec²(a)
- 1 + cot²(a) = csc²(a)
Now we arrive at:
- 5 + sec²(a) + csc²(a)
Next, using computed values according to specific ‘a’ values, you would substitute values into the final formula derived:
- For example, if a = 45°, tan(45°) = 1 and cot(45°) = 1, leading to:
- 5 + 1 + 1 = 7
Therefore, the expression evaluates specifically based on the input value of ‘a’. Always consider what angle you are working with to yield a final computational result.