(tan/cot)- (sec/ cos)
Also I need help with tan sin +cos = sec
retype in proper form.
Are we solving or proving an identity?
tan = sin(x)/cos(x)
cot = cos(x)/sin(x)
sec = 1/cos(x)
(sin(x))^2 + (cos(x))^2 = 1
tan(x)/cot(x) - sec(x)/cos(x)
= (sin(x))^2/(cos(x))^2 - 1/(cos(x))^2
= ((sin(x))^2-1)/(cos(x))^2
= (cos(x))^2/(cos(x)^2) = 1
tan(x).sin(x) + cos(x)
= (sin(x))^2/cos(x) + cos(x)
= ((sin(x))^2 + (cos(x))^2)/cos(x)
= 1/cos(x) = sec(x)
Hope this helps!
Proving it!(:
To simplify the expression (tan/cot) - (sec/cos), we need to understand the trigonometric identities.
1. tan/cot = sin/cos : This identity states that the tangent of an angle is equal to the sine of the angle divided by the cosine of the angle. Similarly, the cotangent of an angle is equal to the cosine of the angle divided by the sine of the angle.
2. sec/cos = 1 : This identity states that the secant of an angle is equal to one divided by the cosine of the angle.
Using these identities, we can rewrite the expression as:
(sin/cos) - 1
To further simplify the expression, we can find a common denominator for sin/cos and 1:
(sin - cos)/cos
So, the simplified expression is (sin - cos)/cos.
Now, let's move on to the second part of your question:
tan(sin) + cos = sec
To solve this equation, we'll use the following trigonometric identity:
tan(x) = sin(x)/cos(x)
sec(x) = 1/cos(x)
Let's substitute these identities into the equation:
(sin/cos) + cos = 1/cos
Now, we can find a common denominator for (sin/cos) and 1/cos:
(sin + cos)/cos = 1/cos
Next, we can cross multiply:
cos * (sin + cos) = 1
Expanding the left side of the equation, we get:
sin*cos + cos^2 = 1
Finally, rearranging the equation, we have:
cos^2 + sin*cos - 1 = 0
This equation can be solved using various methods such as factoring, quadratic formula, or trigonometric identities.