Simplify the expression sin²x(1+cot²x)
simplify the trigonometry expression by showing working out
sin x/cos x + cos x /sin x
To simplify the expression sin²x(1+cot²x), we can first rewrite cot²x in terms of sin²x.
Recall that cot(x) = cos(x)/sin(x).
Substituting this into the expression, we have:
sin²x(1 + (cos²x/sin²x))
Next, simplify the expression by combining like terms:
sin²x(1 + cos²x/sin²x)
To simplify further, we can multiply sin²x by 1 to get:
sin²x + sin²x(cos²x/sin²x)
Simplifying the expression, we have:
sin²x + cos²x
This can be further simplified using the trigonometric identity cos²x = 1 - sin²x:
sin²x + (1 - sin²x)
Finally, combining the terms, we have:
1
To simplify the expression sin²x(1+cot²x), we'll first use the identity cot²x = 1/tan²x.
Substituting this into the expression, we get sin²x(1+1/tan²x).
Next, we'll use another identity, tan²x = sec²x - 1.
By substituting this identity into the expression, we have sin²x(1+1/(sec²x - 1)).
Now, let's simplify further.
To get a common denominator, we need to multiply the second term 1/(sec²x - 1) by (sec²x + 1)/(sec²x + 1):
sin²x(1+1/(sec²x - 1)) = sin²x(1*(sec²x + 1)/(sec²x - 1) + 1/(sec²x - 1)).
Expanding the expression and combining like terms:
sin²x(sec²x + 1)/(sec²x - 1) + sin²x/(sec²x - 1).
Next, we'll simplify the denominator by using the identity sec²x - 1 = tan²x:
sin²x(sec²x + 1)/(tan²x) + sin²x/(tan²x).
Multiplying sin²x by sec²x + 1 in the numerator:
sin²x(sec²x + 1)/(tan²x) + sin²x/(tan²x).
Finally, to simplify further, we can combine the terms by finding a common denominator:
[sin²x(sec²x + 1) + sin²x]/(tan²x).
Now, let's simplify the numerator by distributing sin²x:
[sin²x * sec²x + sin²x + sin²x]/(tan²x).
Combining like terms in the numerator:
[sin²x * sec²x + 2sin²x]/(tan²x).
And, this is the simplified expression for sin²x(1+cot²x).