If-x-cos-sin-and-y-cos2-Show-that-y-x-2-x-2-

Question Number 7608 by Tawakalitu. last updated on 06/Sep/16

I f x = c o s Θ - s i n Θ

a n d y = c o s 2 Θ

S h o w t h a t,

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

Commented by sou1618 last updated on 06/Sep/16

\sqrt{2 - x^{2}} = \sqrt{1 + 2 c o s θ s i n θ}

=∣ c o s θ + s i n θ ∣

s o

x \sqrt{2 - x} = (c o s θ - s i n θ) ∣ c o s θ + s i n θ ∣

y = c o s 2 θ = {c o s}^{2} θ - {s i n}^{2} θ

= (c o s θ - s i n θ) (c o s θ + s i n θ)

w h e n (c o s θ + s i n θ) ⩾ 0

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

w h e n (c o s θ + s i n θ) < 0

y \neq x \sqrt{2 - x^{2}}

Commented by Tawakalitu. last updated on 06/Sep/16

T h a n k y o u s i r

Commented by Tawakalitu. last updated on 06/Sep/16

i a p p r e c i a t e

Answered by sandy_suhendra last updated on 06/Sep/16

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

= (c o s θ - s i n θ) \sqrt{2 ({c o s}^{2} θ + {s i n}^{2} θ) - {(c o s θ - s i n θ)}^{2}}

= (c o s θ - s i n θ) \sqrt{2 {c o s}^{2} θ + 2 {s i n}^{2} θ) - ({c o s}^{2} θ + {s i n}^{2} θ - 2 s i n θ c o s θ})

= (c o s θ - s i n θ) \sqrt{{c o s}^{2} θ + {s i n}^{2} θ + 2 s i n θ c o s θ}

= (c o s θ - s i n θ) \sqrt{{(c o s θ + s i n θ)}^{2}}

= (c o s θ - s i n θ) (c o s θ + s i n θ)

= {c o s}^{2} θ - {s i n}^{2} θ

= c o s 2 θ

= y

Commented by Tawakalitu. last updated on 06/Sep/16

i r e a l l y a p p r e c i a t e y o u r e f f o r t s i r . t h a n k s s o m u c h .

Commented by sou1618 last updated on 06/Sep/16

\sqrt{{(c o s θ + s i n θ)}^{2}} - (c o s θ + s i n θ) \overset{? ?}{=} 0

i f θ = 0

\sqrt{{(1 + 0)}^{2}} - (1 + 0) = 0

i f θ = π

\sqrt{{(0 - 1)}^{2}} - (0 - 1) = 2

I t h i n k \sqrt{a^{2}} =∣ a ∣

e . g .

a = - 2

\sqrt{{(- 2)}^{2}} = 2

Commented by Rasheed Soomro last updated on 07/Sep/16

I a g r e e w i t h y o u .