Question-205001

Question Number 205001 by rajusasmal last updated on 05/Mar/24

Answered by TonyCWX08 last updated on 05/Mar/24

44 .

x^{2} - y^{2}

= {(a s e c (θ) + b t a n (θ))}^{2} - {(a t a n (θ) + b s e c (θ))}^{2}

= a^{2} {s e c}^{2} (θ) + 2 a b s e c (θ) t a n (θ) + b^{2} {t a n}^{2} (θ) - a^{2} {t a n}^{2} (θ) - 2 a b t a n (θ) s e c (θ) - b^{2} {s e c}^{2} (θ)

= a^{2} {s e c}^{2} (θ) - a^{2} {t a n}^{2} (θ) + b^{2} {t a n}^{2} (θ) - b^{2} {s e c}^{2} (θ)

= a^{2} ({s e c}^{2} (θ) - {t a n}^{2} (θ)) + b^{2} ({t a n}^{2} (θ) - {s e c}^{2} (θ))

= a^{2} (1) + b^{2} (- 1)

= a^{2} - b^{2}

45 .

3 s i n (x) + 5 c o s (x) = 5

3 {s i n}^{2} (x) + 30 s i n (x) c o s (x) + 25 {c o s}^{2} (x) = 25

30 s i n (x) c o s (x) = 25 - 3 {s i n}^{2} (x) - 25 {c o s}^{2} (x)

{(5 s i n (x) - 3 c o s (x))}^{2}

= 25 {s i n}^{2} (x) - 30 s i n (x) c o s (x) + 9 {c o s}^{2} (x)

= 25 {s i n}^{2} (x) - (25 - 3 {s i n}^{2} (x) - 25 {c o s}^{2} (x)) + 9 {c o s}^{2} (x)

= 25 {s i n}^{2} (x) - 25 + 3 {s i n}^{2} (x) + 25 {c o s}^{2} (x) + 9 {c o s}^{2} (x)

= 34 {s i n}^{2} (x) + 34 {c o s}^{2} (x) - 25

= 34 ({s i n}^{2} (x) + {c o s}^{2} (x)) - 25

= 34 - 25

= 9

5 s i n (x) - 3 c o s (x) = \pm \sqrt{9} = \pm 3

Answered by TonyCWX08 last updated on 05/Mar/24

36 .

x^{2} + y^{2} + z^{2}

= r^{2} {s i n}^{2} (A) {c o s}^{2} (C) + r^{2} {s i n}^{2} (A) {s i n}^{2} (C) + r^{2} {c o s}^{2} (A)

= r^{2} ({s i n}^{2} (A) ({s i n}^{2} (C) + {c o s}^{2} (C)) + {c o s}^{2} (A))

= r^{2} ({s i n}^{2} (A) + {c o s}^{2} (A))

= r^{2} (1)

= r^{2}

40 .

s i n (x) + {s i n}^{2} (x) = 1

{s i n}^{2} (x) = 1 - s i n (x)

1 - {c o s}^{2} (x) = 1 - s i n (x)

- {c o s}^{2} (x) = - s i n (x)

{c o s}^{2} (x) = s i n (x)

{c o s}^{4} (x) = {s i n}^{2} (x)

{c o s}^{2} (x) + {c o s}^{4} (x)

= s i n (x) + {s i n}^{2} (x) = 1