Verify the identity.
cos x - sec x = -sin x tan x
So far I have applied a reciprocal identity
=cos x - 1/cos x
I know I must combine using the least common denominator but I don't know how it would be written.
multiplying by cosine
... cos^2(x) - 1 = sin^2(x)
To prove an identity, we usually start with one side and using basic identities show that it is equal to the other side
LS = cosx - sex
= cosx - 1/cosx
= (cos^2 x - 1)/cosx
= - sin^2 x / cosx
= -sinx(sinx/cosx)
= -sinx tanx
= RS
To verify the given trigonometric identity, you correctly applied the reciprocal identity for secant:
cos x - sec x = cos x - 1/cos x
To simplify this expression by combining the terms using the least common denominator, we need to first find the least common denominator (LCD) between cos x and 1/cos x.
The LCD is determined by finding the common factors and multiplying them together. In this case, the LCD is cos x.
Now, we can rewrite the expression using the LCD:
cos x - 1/cos x = (cos x * cos x - 1) / cos x
Next, we can simplify the numerator:
cos^2 x - 1 = (cos x - 1)(cos x + 1)
Now, let's simplify the expression further. We know that:
cos^2 x + sin^2 x = 1
Rearranging this equation, we have:
cos^2 x = 1 - sin^2 x
Substituting this into our simplified expression:
(cos x - 1)(cos x + 1) = (1 - sin^2 x - 1)
The "1" terms cancel out, leaving us with:
(cos x - 1)(cos x + 1) = -sin^2 x
Now, we can rewrite the right side of the equation using the Pythagorean Identity:
-sin^2 x = -1 * (1 - cos^2 x) = -1 + cos^2 x
Therefore, we have:
(cos x - 1)(cos x + 1) = -1 + cos^2 x
Finally, we can simplify the left side of the equation:
(cos x - 1)(cos x + 1) = cos^2 x - 1
And the right side of the equation:
-1 + cos^2 x = cos^2 x - 1
Hence, the left and right sides of the equation are equal, which verifies the given trigonometric identity:
cos x - sec x = -sin x tan x