Write the following trigonometric expression in terms of sine and cosine, and then simplify. sin^2 x (1 + cot^2 x)
well, cotx = cosx/sinx, so you have
sin^2x + sin^2x cot^2x = sin^2x + cos^2x
Look familiar?
To express the trigonometric expression sin^2(x)(1 + cot^2(x)) in terms of sine and cosine, we can use the reciprocal identities:
cot(x) = cos(x) / sin(x),
sin^2(x) = 1 - cos^2(x).
Substituting these identities into the expression, we get:
sin^2(x)(1 + cot^2(x)) = sin^2(x)(1 + (cos(x) / sin(x))^2).
Simplifying further using basic algebra, we have:
sin^2(x)(1 + cot^2(x)) = sin^2(x)(1 + cos^2(x) / sin^2(x)).
Now, we can simplify the expression by canceling out the sin^2(x) terms:
sin^2(x)(1 + cos^2(x) / sin^2(x)) = sin^2(x) + cos^2(x).
Finally, using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we find:
sin^2(x)(1 + cot^2(x)) = 1.
Therefore, the simplified form of the trigonometric expression sin^2(x)(1 + cot^2(x)) is 1.
To write the trigonometric expression sin^2 x (1 + cot^2 x) in terms of sine and cosine, we can use the relationships between trigonometric functions.
Recall the following relationships:
- cot x = cos x / sin x
- cot^2 x = (cos x / sin x)^2 = cos^2 x / sin^2 x
- sin^2 x = 1 - cos^2 x
Let's break down the expression step by step:
sin^2 x (1 + cot^2 x)
Using the relationship cot^2 x = cos^2 x / sin^2 x:
sin^2 x (1 + cos^2 x / sin^2 x)
Simplifying the expression inside the parentheses:
sin^2 x (sin^2 x / sin^2 x + cos^2 x / sin^2 x)
Combining the fractions inside the parentheses:
sin^2 x ((sin^2 x + cos^2 x) / sin^2 x)
Since sin^2 x + cos^2 x = 1, the expression simplifies to:
sin^2 x (1 / sin^2 x)
Simplifying further:
1
Therefore, the final simplified expression is 1.