For each of the following determine whether or not it is an identity and prove your result.
a. cos(x)sec(x)-sin^2(x)=cos^2(x)
b. tan(x+(pi/4))= (tan(x)+1)/(1-tan(x))
c. (cos(x+y))/(cos(x-y))= (1-tan(x)tan(y))/(1+tan(x)tan(y))
d. (tan(x)+sin(x))/(1+cos(x))=tan(x)
e. (sin(x-y))/(sin(x)cos(y))=1-cot(x)tan(y)
f. sin(x)sin(y)=(1/2)[cos(x-y)-cos(x+y)]
a
1 - s^2 = c^2
yes
s^2+c^2 = 1
b
t(x+pi/4) = (t x + 1)/(1-t x)
yes
c
t x t y = s x /c x * s y/c y
so on the right we have
[1 - (s x s y / c x c y) ] / [ 1 + ( s x s y /c x c y) ]
[c x c y - s x s y] / [c x c y + s x s y]
which is indeed
c(x+y)/c(x-y)
so yes
d I am getting bored. I think you cn see the plan.
To determine whether each of the given equations is an identity, we need to prove that both sides of the equation are equal for all values of the variable(s) involved. Let's go through each equation and prove the results.
a. cos(x)sec(x) - sin^2(x) = cos^2(x)
First, we will simplify the left side of the equation using trigonometric identities:
cos(x)sec(x) - sin^2(x)
= cos(x)(1/cos(x)) - sin^2(x)
= 1 - sin^2(x) (using the identity sec(x) = 1/cos(x))
= cos^2(x) (using the identity 1 - sin^2(x) = cos^2(x))
Since the left side simplifies to the right side, the equation is an identity.
b. tan(x + π/4) = (tan(x) + 1) / (1 - tan(x))
First, let's simplify the left side of the equation using the angle addition formula:
tan(x + π/4)
= (tan(x) + tan(π/4)) / (1 - tan(x)tan(π/4))
= (tan(x) + 1) / (1 - tan(x)) (since tan(π/4) = 1)
Since the left side simplifies to the right side, the equation is an identity.
c. (cos(x+y))/(cos(x-y)) = (1 - tan(x)tan(y))/(1 + tan(x)tan(y))
To prove this equation, we need to start by using the double-angle formula for cosine:
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
Now, let's substitute these values into the original equation:
(cos(x)cos(y) - sin(x)sin(y)) / (cos(x)cos(y) + sin(x)sin(y))
= ((cos(x)cos(y) - sin(x)sin(y))/(cos(x)cos(y))) / ((cos(x)cos(y) + sin(x)sin(y))/(cos(x)cos(y)))
= (cos(x) - tan(x)tan(y))/(cos(x) + tan(x)tan(y)) (dividing numerator and denominator by cos(x)cos(y))
Since both sides of the equation simplify to the same expression, the equation is an identity.
d. (tan(x) + sin(x))/(1 + cos(x)) = tan(x)
Let's simplify the left side of the equation using trigonometric identities:
(tan(x) + sin(x))/(1 + cos(x))
= (sin(x)/cos(x) + sin(x))/(1 + cos(x))
= sin(x)(1 + cos(x))/(cos(x)(1 + cos(x)))
= sin(x)/cos(x) (canceling out (1 + cos(x)) terms)
Since sin(x)/cos(x) is equal to tan(x), the equation is an identity.
e. (sin(x-y))/(sin(x)cos(y)) = 1 - cot(x)tan(y)
To prove this equation, we need to use the quotient identities and simplify both sides:
(siny*cosy*cosx - cosy*sinx)/(sinx*cosy)
= sinx*cosy*cosx/(sinx*cosy) - cosy*sinx/(sinx*cosy)
= cosx - cotx*sinx (canceling out sinx*cosy terms)
Since both sides of the equation simplify to the same expression, the equation is an identity.
f. sin(x)sin(y) = (1/2)(cos(x-y) - cos(x+y))
To prove this equation, we need to use the angle subtraction and addition formulas for cosine:
(1/2)(cos(x-y) - cos(x+y))
= (1/2)((cos(x)cos(y) + sin(x)sin(y)) - (cos(x)cos(y) - sin(x)sin(y)))
= (1/2)(2sin(x)sin(y))
= sin(x)sin(y)
Since both sides of the equation are equal, the equation is an identity.
Therefore, we have proved that all the given equations are identities.