verify sinx/secx = 1/tanx+cotx
sinx/secx = 1/tanx+cotx , is not true but
sinx/secx = 1/(tanx+cotx) is true
LS = sinx(1/cosx)
=sinxcosx
RS = 1/(sinx/cosx + cosx/sinx)
= 1/( (sin^2 x + cos^2 x)/(sinxcosx) )
= 1/(1/sinxcosx))
= sinxcosx
= LS
To verify the given equation sin(x)/sec(x) = 1/tan(x) + cot(x), we need to simplify both sides of the equation and check if they are equal.
Let's start by simplifying the left side of the equation:
sin(x)/sec(x)
Using the reciprocal identity, sec(x) = 1/cos(x), we can rewrite sec(x) as 1/cos(x):
sin(x) / (1/cos(x))
Next, we can multiply sin(x) by cos(x) to get rid of the fraction:
sin(x) * cos(x) / 1
Using the product-to-sum identity, sin(x) * cos(x) is equal to 1/2 * [sin(2x)]:
(1/2) * sin(2x) / 1
Simplifying further, we have:
(1/2) * sin(2x)
Now, let's simplify the right side of the equation:
1/tan(x) + cot(x)
Using the definitions of tan(x) and cot(x), we have:
1/(sin(x)/cos(x)) + cos(x)/sin(x)
To add these two fractions, we need a common denominator. The common denominator is sin(x)*cos(x):
(cos(x) + cos^2(x)) / (sin(x)*cos(x))
Using the fact that cos^2(x) - sin^2(x) = cos(2x), we can rewrite the numerator:
cos(2x)
Now that we have simplified both sides of the equation, we can compare them:
(1/2) * sin(2x) = cos(2x)
The left side is equal to the right side: sin(2x) = 2sin(x)cos(x), which is a known identity.
Therefore, we have verified that sin(x)/sec(x) = 1/tan(x) + cot(x).