Question

Verify the following identity:

Answer 1

Answer:

$\rm 2 \sec {}^{2} T$(Verified identity)

Step-by-step explanation:

Please refer to the attachment for calculations.

I hope this helps!

Answer 2

We will simplify the left hand side of the equation to look like the right

$\displaystyle \frac{1-sin(t)}{cos^{2}(t)} + \frac{1}{1 - sin(t)}$

taking the LCM

$\displaystyle \frac{[1-sin(t)]^{2} + cos^{2}(t)}{[cos^{2}(t)][1 - sin(t)]}$

expanding the binomial in the numerator

$\displaystyle \frac{[1 + sin^{2}(t) - 2sin(t) + cos^{2}(t)]}{[cos^{2}(t)][1 - sin(t)]}$

Since sin²x + cos²x = 1

$\displaystyle \frac{[2 - 2sin(t)]}{[cos^{2}(t)][1 - sin(t)]}$

factoring out the 2 from the numerator

$\displaystyle \frac{2[1 - sin(t)]}{[cos^{2}(t)][1 - sin(t)]}$

1-sin(t) will cancel out

$\displaystyle \frac{2}{cos^{2}(t)}$

since cos²(t) = 1/(sec²(t))

$2 sec^{2}(t)$

which is equal to the right hand side of our given equation.

So we verified the identity!