Prove that tan2(x) + 1 = sec2(x)

Prove that tan²(x) + 1 = sec²(x)

We know that

sin²(x) + cos²(x) = 1

dividing both sides by cos²(x) we get

(sin²(x) + cos²(x))/cos²(x) = 1/cos²(x)

which equals

sin²(x)/cos²(x) + cos²(x)/cos²(x) = (1/cos(x))²

which can be re-written as

(sin(x)/(cos(x))² + 1 = (1/cos(x))²


We know that tan²(x) = (sin(x)/(cos(x))²

and that (1/cos(x))² = sec²(x)

So we can write

cot²(x) + 1 = sec²(x)

Prove that tan(x) = sin(x)/cos(x)

Prove that tan(x) = sin(x)/cos(x)



sin(x) = (0/H)

cos(x) = (A/H)

Then sin(x)/cos(x) =

(O/H)/(A/H) =

(O/H)*(H/A) =

(O/A) = tan(x)

So sin(x)/cos(x) = tan(x)

Tangent

The tangent function is defined as the length of the side opposite the angle x divided by the length of the adjacent angle.



Thus if O=5 and A=3 then

tan(x)=5/3