simplify (sec^2x + csc^2x) - (tan^x +cot^2x) to either a constant or a basic trigonometric function.
To simplify the given expression: (sec^2x + csc^2x) - (tan^x + cot^2x)
We'll simplify it step by step:
1. Let's simplify sec^2x + csc^2x:
Recall the trigonometric identity:
sec^2x = 1 + tan^2x
Substituting this into the expression:
sec^2x + csc^2x = (1 + tan^2x) + csc^2x
Now, let's use another trigonometric identity:
csc^2x = 1 + cot^2x
Substituting this back into the expression:
(1 + tan^2x) + (1 + cot^2x) = 1 + tan^2x + 1 + cot^2x
Simplifying this further, we get:
2 + tan^2x + cot^2x
2. Now, let's simplify the second part of the expression, tan^x + cot^2x:
Recall the trigonometric identity:
cot^2x = 1 + tan^2x
Substituting this into the expression:
tan^x + cot^2x = tan^x + (1 + tan^2x)
Simplifying this further, we get:
1 + tan^x + tan^2x
3. Finally, substitute the resulting expressions back into the original expression:
(2 + tan^2x + cot^2x) - (1 + tan^x + tan^2x)
Simplify further by combining like terms:
2 - 1 + tan^2x - tan^2x + cot^2x - tan^x
1 + cot^2x - tan^x
So, the simplified expression is 1 + cot^2x - tan^x.