For each trigonometric expression A,B,C,D, E, choose the expression from 1,2,3,4,5 that completes a fundamental identity. Enter the appropriate letter (A,B,C,D, or E) in each blank.

Question

1. 1
2. 1-cos^2(x)
3. cot(x)
4. sec^2(x)
5. tan(x)

A. sin^2(x)

B. (cos(x)/sin(x))

C. sin^2(x)+cos^2(x)

D. (sin(x)/cos(x))

E. tan^2(x)+1

Answer 1

I would recommend looking up the magic trig hexagon, it has all of these identities and more within it.

1. 1 - this corresponds with C as sin^2(x)+cos^2(x)=1
2. 1-cos^2(x) - this corresponds with A, using the identity from number 1, we can rewrite it in the form sin^2(x)=1-cos^2(x)
3. cot(x) - for this it is important to know that cotangent is the inverse of tangent. Since tan(x)=sin(x)/cos(x), cot=cos(x)/sin(x) which is B.
4. sec^2(x) - much like the cos and sin pythagorean identity, sec and tan are related. sec^2(x)=tan^2(x)+1 which is answer choice E.
5. tan(x) - this is sin(x)/cos(x), choice D.

Answer 2

You need to remember the trig identity sin^2 x + cos^2 x=1 We can subtract cos^2 x from both sides of the equation we get sin^2 x = 1- cos^2 x

So 1 matches C and 2 matches C

Tangent is sin/cos and cotangent is cos/sin These are basic trig identities you need to know

So 3 matches B an and 5 matches D

Lets go back to sin^2 x + cos^2 x =1 Divide both sides of the equation by cos^ x This gives us

sin^2 x/cos^2 x + cos^2 x/ cos^2 x = 1/cos^2 x

tangent^2 x + 1 = secant^2 x

So 4 matches up with E

Let me know if anything here is not clear