Prove csc(pi/2-x)=sec x.
a. csc(pi/2-x)=1/sin(pi/2)cosx+cos(pi/2)cosx=sec x
b. csc(pi/2-x)=1/sin(pi/2)sinx-cos(pi/2)cosx=sec x
c. csc(pi/2-x)=1/sin(pi/2)sinx+cos(pi/2)cosx=sec x
d. csc(pi/2-x)=1/sin(pi/2)cosx-cos(pi/2)sinx=sec x
preliminary discussion:
complementary angles are a pair of angles that add up to 90°
btw, π/2 radians = 90°
there are 3 pairs of trig functions that are co-functions:
sine vs cosine
tangent vs cotangent
secant vs cosecant
e.g. sin20° = cos 70° , tan 28° = cot 62° , sec 10.2 = csc 79.8°
try them on your calculator.
I guess we are to prove one of them:
csc(pi/2-x)=sec x
LS =1/(sin(π/2) - x)
= 1/((sin(π/2)cosx - cos(π/2)sinx)
= 1/( 1(cosx - 0(sinx))
= 1/cosx
= sec x
= RS
A 5 line proof.
I have no idea what your choices are supposed to be.
In short, it's D. csc(pi/2-x)=1/sin pi/2 cos x-cos pi/2 sin x=sec x
It's on brainly too, so check it if you don't believe me. You're welcome, have a nice day <3
To prove that csc(pi/2 - x) = sec(x), we need to manipulate the left side of the equation until it becomes equivalent to the right side (sec(x)). Let's go through the options one by one to determine which one is correct:
a. csc(pi/2 - x) = 1/(sin(pi/2)cos(x) + cos(pi/2)cos(x))
This option is incorrect because it involves two positive terms in the denominator, which doesn't match the structure of sec(x).
b. csc(pi/2 - x) = 1/(sin(pi/2)sin(x) - cos(pi/2)cos(x))
Similarly, this option is incorrect because the denominator consists of one positive term and one negative term, not matching the structure of sec(x).
c. csc(pi/2 - x) = 1/(sin(pi/2)sin(x) + cos(pi/2)cos(x))
This is the correct option! By using the Pythagorean Identity (sin^2(x) + cos^2(x) = 1), we can simplify the denominator of this expression:
sin(pi/2)sin(x) + cos(pi/2)cos(x) = cos(x)sin(x) + 0 = cos(x)sin(x)
Thus, csc(pi/2 - x) = 1/(cos(x)sin(x)) = sec(x), which proves the given equation.
d. csc(pi/2 - x) = 1/(sin(pi/2)cos(x) - cos(pi/2)sin(x))
This option is incorrect because the numerator consists of one positive term and one negative term, unlike the structure of sec(x).
Therefore, the correct answer is option c. csc(pi/2 - x) = 1/(sin(pi/2)sin(x) + cos(pi/2)cos(x)) = sec(x).
the co- means complement.
so, cosec(x) = sec(x-complement) = sec(pi/2-x)
and so forth
If you play around with a right triangle, where the two acute angles are complementary, you can see how it all works.