trigonometric ratios and identities Model Questions & Answers, Practice Test for ssc cgl tier 1 2024

Answer: (a)

Given ${cos^2 θ \text"- 3 cos θ + 2"}/{sin^2 θ}$ = 1

⇒ ${(\text"2 - cos θ") (\text"1 - cos θ")}/{1 - cos^2 θ}$ = 1

⇒ ${\text"2 - cos θ"}/{\text"1 + cos θ"}$ = 1

⇒ 2 - cos θ = 1 + cos θ

⇒ cos θ = $1/2$ = cos 60°

⇒ θ = 60° As 0 < θ < ${π}/2$

There is only one value of θ satisfying the above equation.

Statement (1) is not correct.

Again put θ = 60° L.H.S of (1)

${cos^2 60° - 3 cos 60° + 2}/{sin^2 60°}$

= ${1/4 - 3/2 + 2}/{3/4} = {1 - 6 + 8}/3$

= $3/3$ = 1 = RHS

Statement (2) is correct.

∴ Option (b) is correct.

Answer: (a)

Statement 1

$1/{\text"1 - sin x"} = 4 + 2 √3$

⇒ 1 = 4 + $2 √3 \text"- 4 sin x - 2" √3$ sin x

⇒ sin x = ${3 + 2 √3}/{4 + 2 √3}$

⇒ sin x = 866 < 1

0 < sin x < 1, Therefore, value of x exists between

0 to ${π}/2$

Statement 2

sin x = $3^{sin^2 x}$

For example x = 45,

sin 45° = 3 sin² 45

⇒ $1/{√2}= 3 (1/{√2})^2 ; 1/{√2} = 3 (1/2)$

So this does not hold good for any values.

So only statement 2 is true.

Answer: (c)

${cos^2 (45° + θ) + cos^2 (45° - θ)}/{tan (60° + θ) tan (30° - θ)}$

= ${{{cos(90° + 2θ)} + 1}/2 + {{cos (90° - 2 θ) + 1}/2}}/{tan (60° + θ) . tan [90° - (60° + θ)]}$

$(∵ \text"cos 2 θ = 2" cos^2$ θ - 1)

= ${{{cos (90° + 2 θ) + cos (90° - 2 θ)} + 1}/2}/{tan (60° + θ) cot (60° + θ)}$

= ${{\text"- sin 2 θ + sin 2 θ"}/2 + 1}/1 = 1$

Answer: (c)

${3 - tan^2 A}/{1 - 3 tan^2 A}$ = K

3 - $tan^2 A = K - 3K tan^2 A$

3K $tan^2 A - tan^2 A$ = K - 3

$tan^2 A$ (3K - 1) = K - 3

$tan^2 A = {K - 3}/{3K - 1}$....(i)

Subject to the condition K > 3 or K < $1/3.$

cosec A $(3 sin A - 4 sin^3 A) = 3 - 4 sin^2 A$

$cot^2 A = {3K - 1}/{K - 3}$

$cosec^2 A = {K - 3 + 3K - 1}/{K - 3} = {4K - 4}/{K - 3}$

$sin^2 A = {K - 3}/{4(K - 1)}$

3 - 4$sin^2 A$ = 3 - ${4(K - 3)}/{4(K - 1)}$

= ${3K - 3 - K + 3}/{K - 1} = {2K}/{K - 1}$

where K > 3 or K < $1/3.$

Answer: (c)

Given that, p° = $q^c$

⇒ $(p. {π}/{180})^c = q^c (∵ 180° = π^c)$

∴ $(p π)^c = (q 180)^c$

∴ π p = 180 q