I'm trying to verify these trigonometric identities.
1. 1 / [sec(x) * tan(x)] = csc(x) - sin(x)
2. csc(x) - sin(x) = cos(x) * cot(x)
3. 1/tan(x) + 1/cot(x) = tan(x) + cot(x)
4. csc(-x)/sec(-x) = -cot(x)
Can you post your attempts?
I will give you a kickstart:
1/(sec(x)tan(x))
=1/(1/cos(x) * sin(x)/cos(x))
=cos²(x)/sin(x)
=(1-sin²(x))/sin(x)
=csc(x) - sin(x)
To verify these trigonometric identities, we can manipulate and simplify both sides of the equation until they are equal.
1. 1 / [sec(x) * tan(x)] = csc(x) - sin(x):
First, let's express the left side of the equation in terms of sine and cosine. We know that sec(x) is equal to 1/cos(x) and tan(x) is equal to sin(x)/cos(x):
1 / [sec(x) * tan(x)] = 1 / [(1/cos(x)) * (sin(x)/cos(x))]
Next, simplify the expression by multiplying the fractions:
1 / [sec(x) * tan(x)] = 1 / [(sin(x))/(cos^2(x))]
To simplify further, multiply the numerator and denominator by cos^2(x):
1 / [sec(x) * tan(x)] = cos^2(x)/sin(x)
Now, we need to manipulate the right side of the equation to try to obtain the same expression. Begin with csc(x) - sin(x):
csc(x) - sin(x) = (1/sin(x)) - sin(x)
To simplify this expression, find a common denominator:
csc(x) - sin(x) = (1 - sin^2(x))/sin(x)
Recall the Pythagorean Identity sin^2(x) + cos^2(x) = 1:
csc(x) - sin(x) = cos^2(x)/sin(x)
We have obtained the expression cos^2(x)/sin(x) on both sides of the equation, so the trigonometric identity is verified.
2. csc(x) - sin(x) = cos(x) * cot(x):
To verify this identity, we can start with the left side of the equation:
csc(x) - sin(x)
Since csc(x) is equal to 1/sin(x), we can rewrite the expression as:
1/sin(x) - sin(x)
To simplify this expression, find a common denominator:
(1 - sin^2(x))/sin(x)
Again, using the Pythagorean Identity sin^2(x) + cos^2(x) = 1:
cos^2(x)/sin(x)
Now, let's look at the right side of the equation, cos(x) * cot(x):
cos(x) * cot(x) = cos(x) * (cos(x)/sin(x))
= cos^2(x)/sin(x)
We have obtained the same expression on both sides of the equation, so the identity is verified.
3. 1/tan(x) + 1/cot(x) = tan(x) + cot(x):
To verify this identity, we can manipulate both sides of the equation separately. Begin with the left side:
1/tan(x) + 1/cot(x)
Recall that tan(x) is equal to sin(x)/cos(x) and cot(x) is equal to cos(x)/sin(x):
1/(sin(x)/cos(x)) + 1/(cos(x)/sin(x))
To simplify this expression, multiply both terms by their reciprocal:
cos(x)/sin(x) + sin(x)/cos(x)
Finding a common denominator:
(cos^2(x) + sin^2(x))/(sin(x) * cos(x))
Using the Pythagorean Identity cos^2(x) + sin^2(x) = 1:
1/(sin(x) * cos(x))
Now, let's look at the right side of the equation, tan(x) + cot(x):
tan(x) + cot(x) = sin(x)/cos(x) + cos(x)/sin(x)
= (sin^2(x) + cos^2(x))/(sin(x) * cos(x))
= 1/(sin(x) * cos(x))
The expression on the right side matches the expression on the left side of the equation, so the identity is verified.
4. csc(-x)/sec(-x) = -cot(x):
Start with the left side of the equation:
csc(-x)/sec(-x)
Recall that csc(x) is equal to 1/sin(x) and sec(x) is equal to 1/cos(x):
(1/sin(-x))/(1/cos(-x))
Sin(-x) is equal to -sin(x) and cos(-x) is equal to cos(x):
(1/(-sin(x)))/(1/cos(x))
To simplify this expression, multiply both numerator and denominator by -1:
(-1/(sin(x)))/(cos(x)/(-1))
This leads to:
(-1/(sin(x)))*(-1/(cos(x)))
Simplifying the expression further yields:
1/(sin(x) * cos(x))
Recall that cot(x) is equal to cos(x)/sin(x):
= -cot(x)
The expression on the right side matches the expression on the left side of the equation, so the identity is verified.