Question

1. Power reducing formulas can be used to rewrite 8sin^2 xcos^2 x as 1-cos4x. True or False?

2. Find the exact value of cos15 degrees using a suitable half angle formula.
square root of 2+root3 / 2?

3. Find the exact value of cos^2(pi/8) using a suitable power reducing formula.
2- root2 / 4, I think..?

4. If secx = -3, and x lies in quadrant II find tan x/2
I got -root2

5. Power reducing formulas can be used to rewrite sin^2 xcos^2 x as 1/8 + cos4x / 4
True?

Answer 1

if a question is simply a True or False, and no work has to be shown, a good method for your #1 would be to take an unusual value of x and test it with your calculator.
e.g. let x = 32.7° (use your calculator memory to store intermediate answers

I get 8sin^2 (32.7°)cos^2 (32.7°) = appr 1.6534..
1 - cos(4(32.7)) = 1.6534
YUP, there is a very high probability that it is TRUE

for #2, your answer > 1, but the cosine of "anything" cannot be > 1 , so you should have known your answer to be incorrect

cos 15° ..... notice 15 is half of 30

cos 30 = 2cos^2 15° - 1
√3/2 + 1 = 2cos^2 15°
(√3 + 2)/4 = cos^2 15°
cos 15° = √(√3+2)/2 , check with a calculator, it works!

4. if secx = -3
then cosx = -1/3 and in II
make a sketch of a triangle, to find
sinx = √8/3
tanx = -√8

so tanx = 2tan(x/2) / (1 - tan^2 (x/2)
let tan (x/2) = t
-√8 = 2t/(1 - t^2)
-√8 + √8t^2 = 2t
√8 t^2 - 2t - √8 = 0
t = (2 ± √(2 + 32)/(2√8)
= 4/√8 or -2/√8
or t = √2 or t = -√2/2 after rationalizing
but if x is in II, the x/2 must be in quadrant I
and tan (x/2) = + √2

hint for #5

sin^2 x cos^2 x
= (sinxcosx)^2
= ( (1/2)sin 2x)^2
= (1/4) sin^2 (2x)

again, why don't you pick a weird angle and test it ?