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).