Sir,
I am having trouble seeing how the following Pythagorean Identities are arrived at in the book..I've tried to use the referenced identities but to no avail...
(a) (tan^2 Theda ) +1 = Secant^2 Theda
On the unit circle recall that y^2 + x^2 = hypotenuse^2 = 1.
By definition tan(theta) = y / x, so
tan^2(theta) + 1 = y^2 / x^2 + 1 = y^2 / x^2 + x^2 / x^2 = (y^2 + x^2) / x^2 = 1 / x^2.
Sec(theta) = 1 / x so the above result is sec^2(theta).
(b) (1+ Cotan^2 Theda)= Cosine^2 Theda
On the unit circle cot(theta) = x / y, so
(1 + cot^2(theta)) = 1 + x^2 / y^2 = y^2 / y^2 + x^2 / y^2 = (y^2 + x^2) / x^2 = 1 / x^2.
This is cosec^2(theta), not cosine^2(theta).