Verify the following:
1. cos x/(1-sinx)= sec x + tan x
2. (tanx+1)^2=sec^2x + 2tan x
3. csc x = )cot x + tan x)/sec x
4. sin2x - cot x = -cotxcos2x
Hint:
change all trig ratios to sines and cosines.
I will do one for you.
3. csc x = (cot x + tan x)/sec x
RS = (cosx/sinx + sinx/cosx)/(1/cosx)
= (cos^2 x + sin^2 x)/(sinxcosx) * cosx
= (1/(sinxcosx)(cosx)
= 1/sinx
= csc x
= LS
To verify each of the given equations, we will simplify the left-hand side (LHS) and the right-hand side (RHS) of each equation separately. Then we will compare the LHS and RHS to see if they are equivalent.
1. Verify cos x/(1-sinx) = sec x + tan x:
To simplify the LHS, we'll multiply the numerator and denominator by (1 + sin x) to get rid of the fraction:
cos x / (1 - sin x) = (cos x * (1 + sin x)) / ((1 - sin x) * (1 + sin x))
= cos x + cos x * sin x / (1 - sin^2 x)
= cos x + cos x * sin x / cos^2 x
= cos x + sin x / cos x
Using the trigonometric identity sin x / cos x = tan x, we can simplify further:
cos x + sin x / cos x = cos x + tan x = RHS
Therefore, the equation is verified.
2. Verify (tanx + 1)^2 = sec^2x + 2tan x:
First, we'll square both sides of the equation:
(tan x + 1)^2 = (sec^2 x + 2tan x)^2
Expanding the squares on both sides:
tan^2 x + 2tan x + 1 = sec^4 x + 4sec^2 x tan x + 4tan^2 x
Using the trigonometric identity sec^2 x = 1 + tan^2 x, we can substitute it into the equation:
tan^2 x + 2tan x + 1 = (1 + tan^2 x)^2 + 4(1 + tan^2 x) tan x + 4tan^2 x
Expanding further:
tan^2 x + 2tan x + 1 = 1 + 2tan^2 x + tan^4 x + 4tan x + 4tan^3 x + 4tan^2 x
Simplifying, regrouping like terms, and canceling out common terms:
tan^4 x + 6tan^2 x + 1 = tan^4 x + 8tan^3 x + 6tan^2 x + 1
Therefore, the equation holds true.
3. Verify csc x = (cot x + tan x)/sec x:
To simplify the RHS, we'll first convert cot x and tan x in terms of sin x and cos x:
cot x = cos x / sin x
tan x = sin x / cos x
Substituting these values into the RHS:
(cot x + tan x) / sec x = (cos x / sin x + sin x / cos x) / (1 / cos x)
= [(cos^2 x + sin^2 x) / (sin x cos x)] / (1 / cos x)
= (1 / (sin x cos x)) / (1 / cos x)
= 1 / sin x
= csc x = LHS
Hence, the equation is verified.
4. Verify sin 2x - cot x = -cot x cos 2x:
Using the double-angle formula for sine (sin 2x = 2sin x cos x) and the reciprocal identity for cotangent (cot x = cos x / sin x):
sin 2x - cot x = 2sin x cos x - (cos x / sin x)
= 2sin^2 x cos x - cos x / sin x
= (2sin^2 x cos x - cos^2 x) / sin x
= cos x (2sin^2 x - cos x) / sin x
= cos x (2(1 - cos^2 x) - cos x) / sin x
= cos x (2 - 2cos^2 x - cos x) / sin x
= (2cos x - 2cos^3 x - cos^2 x) / sin x
Using the Pythagorean identity sin^2 x + cos^2 x = 1:
= (2cos x - 2cos^3 x - (1 - sin^2 x)) / sin x
= (2cos x - 2cos^3 x - 1 + sin^2 x) / sin x
= (2cos x - 2cos^3 x - 1 + 1 - cos^2 x) / sin x
= (cos x - cos^3 x - cos^2 x) / sin x
= - cot x cos^2 x / sin x
= - cot x cos x (cos x / sin x)
= - cot x cos 2x = RHS
Therefore, the equation is verified.