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SOH, CAH, TOA explained using an example. 

Let us procede by induction:

First case: n=1. Then LHS (left hand side) = 1 and RHS (right hand side) = 1(1+1)/2 = 1. Therefore, we see that the statement is true for n=1.

Now, we carry out ...

x = ln (sec (2y))

The chain rule states that d/dy f (g (y)) = f'(g(y)). g'(y)

Here g(y) = sec(2y) so g'(y) = 2.sec(2y).tan(2y)

And f(y) = ln (y) so f'(y) = 1 / y

Thus dx/dy = (...

x² - 8x + 5 = 0 => (x² - 8x + 16) - 11 = 0 =>(x - 4)² = 11 =>x - 4 = +- sqrt(11) =>x = 4 +- sqrt(11)

k=3