(cotx)(cscx)+secx=(csc^2x)(secx)
Please help to verify this trigonometric identity.
working with the left side,
cos/sin * 1/sin + 1/cos
cos/sin^2 + 1/cos
(cos^2+sin^2)/(sin^2 cos)
1/(sin^2 cos)
1/sin^2 * 1/cos
csc^2 sec
thank you!
you're welcome.
I always enjoyed doing trig identities. They provide endless variations. The hard ones for me, though, are the ones involving tan and sec combinations. They can get quite involved.
cot^2 X - tan^2 X + sec^2 X = csc^2
To verify the given trigonometric identity, we need to simplify both sides of the equation and demonstrate that they are equal. Let's work on it step by step:
Starting with the left side of the equation:
(cot x)(csc x) + sec x
First, let's rewrite the trigonometric functions in terms of sine and cosine:
cot x = cos x / sin x
csc x = 1 / sin x
sec x = 1 / cos x
Substituting these values into the equation, we have:
(cos x / sin x)(1 / sin x) + 1 / cos x
To simplify, we need a common denominator. The common denominator in this case will be sin x * cos x.
Thus, we can rewrite the equation as follows:
[(cos x * cos x) + (sin x * sin x)] / (sin x * cos x)
Using the Pythagorean identity sin^2 x + cos^2 x = 1, we can simplify the numerator:
[1] / (sin x * cos x)
Now, let's simplify the right side of the equation:
(csc^2 x)(sec x)
Substituting the values, we have:
[(1 / sin x)^2](1 / cos x)
(1 / sin^2 x)(1 / cos x)
Using the property of reciprocal of a fraction, we can rewrite the equation as:
1 / [sin^2 x * cos x]
Comparing the left side and right side, we can see that they are indeed equal:
[1 / (sin x * cos x)] = 1 / [sin^2 x * cos x]
Hence, the given trigonometric identity (cot x)(csc x) + sec x = (csc^2 x)(sec x) is verified.