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Question Number 63426 by Rio Michael last updated on 04/Jul/19

s h o w t h a t

(a) c o s [2 {c o s}^{- 1} (x) + {s i n}^{- 1} (x)] = - \sqrt{1 - x^{2}}

(b) \frac{s i n α + s i n β}{c o s α - c o s β} = c o t (\frac{β - α}{2})

(c) 2 c o s (\frac{π}{3} + p) ≊ 1 - \sqrt{3} i f p i s s m a l l e n o u g h t o n e g l e c t p^{2} .

(d) i f θ = \frac{1}{2} {s i n}^{- 1} (\frac{3}{4}), s h o w t h a t s i n θ - c o s θ = \pm \frac{1}{2}

(e) w r i t e t a n 3 A i n t e r m s o f t a n A

(f) F a c t o r i s e c o s θ - c o s 3 θ - c o s 5 θ + c o s 7 θ

(g) i) v e r i f y t h a t f (x) = \frac{s i n 2 θ + s i n 10 θ}{c o s 2 θ + c o s 10 θ} = \frac{2 t a n 3 θ}{1 - {t a n}^{2} 3 θ}

i i) h e n c e f i n d i n r a d i a n s t h e g e n e r a l s o l u t i o n o f f (x) = 1

s i r F o r k u m M i c h a e l

Commented by Tony Lin last updated on 04/Jul/19

(a)

l e t θ = {c o s}^{- 1} (x) \Rightarrow c o s θ = x

l e t \frac{π}{2} - θ = {s i n}^{- 1} (x) \Rightarrow s i n θ = \sqrt{1 - x^{2}}

c o s [2 θ + (\frac{π}{2} - θ)]

= c o s (\frac{π}{2} + θ)

= - s i n θ

= - \sqrt{1 - x^{2}}

Commented by Tony Lin last updated on 04/Jul/19

(b)

\frac{s i n α + s i n β}{c o s α - c o s β}

= \frac{2 s i n (\frac{α + β}{2}) c o s (\frac{α - β}{2})}{- 2 s i n (\frac{α + β}{2}) s i n (\frac{α - β}{2})}

= \frac{- c o s (\frac{α - β}{2})}{s i n (\frac{α - β}{2})}

= c o t (\frac{β - α}{2})

Commented by Tony Lin last updated on 04/Jul/19

(d)

θ = \frac{1}{2} {s i n}^{- 1} (\frac{3}{4})

2 θ = {s i n}^{- 1} (\frac{3}{4})

\Rightarrow \frac{3}{4} = s i n 2 θ

\Rightarrow \frac{3}{4} = 2 s i n θ c o s θ

{(s i n θ - c o s θ)}^{2}

= 1 - 2 s i n θ c o s θ

= \frac{1}{4}

\Rightarrow s i n θ - c o s θ = \pm \frac{1}{2}

Commented by Tony Lin last updated on 04/Jul/19

(e)

t a n 3 A

= t a n (A + 2 A)

= \frac{t a n A + t a n 2 A}{1 - t a n A t a n 2 A}

= \frac{t a n A + \frac{2 t a n A}{1 - {t a n}^{2} A}}{1 - \frac{2 {t a n}^{2} A}{1 - {t a n}^{2} A}}

= \frac{3 t a n A - {t a n}^{3} A}{1 - 3 {t a n}^{2} A}

Commented by Rio Michael last updated on 05/Jul/19

p l e a s e e x p l a i n t h e o c c u r a n c e o f

{(s i n θ - c o s θ)}^{2}

Answered by som(math1967) last updated on 04/Jul/19

g (i) \frac{s i n 2 θ + s i n 10 θ}{c o s 2 θ + c o s 10 θ} = \frac{2 s i n 6 θ c o s 4 θ}{2 c o s 6 θ c o s 4 θ} = t a n 6 θ

a g a i n t a n 6 θ = t a n 2.3 θ = \frac{2 t a n 3 θ}{1 - {t a n}^{2} 3 θ}

i i) A g a i n \frac{2 t a n 3 θ}{1 - {t a n}^{2} 3 θ} = 1

\Rightarrow {t a n}^{2} 3 θ + 2 t a n 3 θ = 1

\Rightarrow {t a n}^{2} 3 θ + 2 t a n 3 θ + 1 = 2

\Rightarrow {(t a n 3 θ + 1)}^{2} = 2

\Rightarrow t a n 3 θ = \sqrt{2} - 1 = t a n \frac{π}{8}

∴ 3 θ = n π + \frac{π}{8} ∴ θ = \frac{1}{3} (n π + \frac{π}{8})