2)Prove the following identities:
a)(cosecx + cotx)(cosecx - cotx) = cotxtanx
b)2/1+sinx + 1/1-sinx = 3sec^2x - tanxcosecx
a) (cscx + cotx)(cscx - cotx)
= csc^2x - cot^2x
= 1/sin^2x - cos^2/sin^2x
= (1-cos^2x)/sin^2x = 1
cotx*tanx also = 1
I will do the second one
2/(1+sinx) + 1/(1-sinx) = 3sec^2x - tanxcosecx
L.S.
= (2(1-sinx) + 1(1+sinx))/(1-sin^2 x)
= (3 - sinx)/cos^2 x
= 3/cos^2 x - sinx/cos^2 x
= 3sec^2 x - (sinx/cos)*1/cosx
= 3sec^2 x - (tanx)(cscx)
= R.S.
thankyou so much !
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truly appreciate the help
To prove the given identities, we'll need to apply basic trigonometric identities and algebraic manipulations. Let's solve each of the identities separately:
a) (cosecx + cotx)(cosecx - cotx) = cotx * tanx
Starting with the left-hand side (LHS):
(cosecx + cotx)(cosecx - cotx) = cosecx * cosecx - cotx * cosecx + cotx * cosecx - cotx * cotx
Now, using the reciprocal identities:
= 1/sinx * 1/sinx - cosx/sinx * 1/sinx + cosx/sinx * 1/sinx - cosx/sinx * cosx/sinx
Simplifying:
= (1 - cosx)/sinx * (1 + cosx)/sinx
Now, applying the product-to-sum identity (a^2 - b^2 = (a+b)(a-b)):
= (1 - cos^2x)/sin^2x
Using the Pythagorean identity (sin^2x + cos^2x = 1):
= sin^2x/sin^2x
Simplifying:
= 1
Hence, LHS = RHS. The identity (cosecx + cotx)(cosecx - cotx) = cotx * tanx is proved.
b) 2/(1 + sinx) + 1/(1 - sinx) = 3sec^2x - tanx * cosecx
Starting with the left-hand side (LHS):
= 2/(1 + sinx) + 1/(1 - sinx)
Now, let's find the common denominator:
= (2(1 - sinx) + 1(1 + sinx))/((1 + sinx)(1 - sinx))
Simplifying:
= (2 - 2sinx + 1 + sinx)/(1 - sin^2x)
Combining like terms:
= (3 - sinx)/(cos^2x)
Using the Pythagorean identity (sin^2x + cos^2x = 1):
= (3 - sinx)/((1 - sin^2x) + sin^2x)
Simplifying:
= (3 - sinx)/1
Simplifying further:
= 3 - sinx
Now let's simplify the right-hand side (RHS):
= 3sec^2x - tanx * cosecx
Since sec^2x = 1/cos^2x and cosecx = 1/sinx:
= 3(1/cos^2x) - (sinx/cosx)(1/sinx)
Simplifying:
= 3/cos^2x - 1/cosx
Getting a common denominator:
= (3 - cosx)/cos^2x
Using the Pythagorean identity (sin^2x + cos^2x = 1):
= (3 - cosx)/(1 - sin^2x + cos^2x)
Simplifying:
= (3 - cosx)/1
Simplifying further:
= 3 - cosx
Hence, LHS = RHS. The identity 2/(1 + sinx) + 1/(1 - sinx) = 3sec^2x - tanx * cosecx is proved.