Some basic trigonometric identities are as follows:

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

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

1 + cot^2(x) = csc^2(x)

Recall that

sin(x) = 1/csc(x)

cos(x) = 1/sec(x)

tan(x) = 1/cot(x)

The trigonometric functions and identities and can derived from the basic trigonometric functions sin(x) and cos(x).

For example:

sin(x) = opposite/hypotenuse,  cos(x) = adjacent/hypotenuse, tan(x) = opposite/adjacent,

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



 1 + tan^2(x) = sec^2(x) using sin(x) and cos(x)

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

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

multiply both sides by cos^2(x) to get

cos^2(x) + sin^2(x) = 1, which confirms the first identity.