Prove the following identities: (i)cos^2x (1+tan^2x)=1, (ii) (2 sin x +cos x)^2=5, (iii) (1+sin x/cos x)^2=1+sin x/1-sin x, (iv) 1- (sin^2 x)/1+cos x= cos x, (v) 1-2cos^2 x/sin x+cos x= sinx-cosx.

Answer 1

When proving an identity work on one side only and that side look like the other side.

For part i).

Working on the left side:

1. convert 1+tan²(x) to sec²(x)
2. convert sec²(x) to 1/cos²(x)
3. Finally cos²(x)/cos²(x) is 1
4. Q.E.D.

Visit the following Desmos graph to see a possible solution for part v.

desmos.com/calculator/5hez7gdu9o

Answer 2

cos^2(x)(1+tan^2(x)) = 1

divide by cos^2(x)

1+tan^2(x) = 1/cos^2(x) = sec^2(x)

1+tan^2(x) = sec^2(x)

multiply by cos^2(x)

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

cos^2(x) + sin^2(x) = 1

which is a version of the Pythagorean Theorem, on a unit circle

converted to rectangular coordinates, it's equivalent to x^2+y^2 = 1

sum of squares of the right triangles side = hypotenuse squared