Question

please answer correctly
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Answer 1

Question 1 :-

• cos 60° = 1/2
• sin 30° = 1/2
• cos 30° = √3/2
• sin 60° = √3/2

$: \implies \sf { \cos 60}^{ \circ} \cdot { \sin60}^{ \circ} - { \cos30 }^{ \circ} \cdot { \sin60}^{ \circ}$

$: \implies \sf \displaystyle \frac{1}{2} \times \frac{1}{2} - \frac{ \sqrt{3} }{2} \times \frac{ \sqrt{3} }{2}$

$: \implies \sf \displaystyle \frac{1}{4} - \frac{3}{4}$

$: \implies \sf \dfrac{ - 2}{4}$

$: \implies \sf \dfrac{ - 1}{2}$

Correct option :- A

Question 2 :-

$\bf \bullet \: { \sec \theta}^{2} - \tan \theta ^{2} = 1$

$: \implies \sf 5 {x}^{2} - \left( \dfrac{5}{x}\right) ^{2} = 1$

$: \implies \sf 25 {x}^{2} - \dfrac{25}{ {x}^{2} } = 1$

$: \implies \sf 25 \times \left( x ^{2} - \dfrac{1}{x} ^{2} \right) = 1$

$: \implies \sf x ^{2} - \dfrac{1}{x} ^{2} = \dfrac{1}{25}$

Correct option :- A

Question 3 :-

$\bf \bullet \: 1 + \tan ^{2} x = \sec ^{2} x$

$: \implies \sf \sin ^{2} x + \dfrac{1}{ \sec ^{2} x}$

$: \implies \sf \sin ^{2} x + \cos ^{2} x$

$: \implies \sf 1$

Correct option :- B

Question 4 :-

$: \implies \sf (1 + \tan \theta + \sec \theta)(1 + \cot \theta - \cosec \theta)$

$: \implies \sf \left(1 + \dfrac{ \sin \theta}{ \cos \theta} + \dfrac{1}{ \cos\theta} \right)\left(1 + \dfrac{ \cos \theta}{ \sin \theta} - \dfrac{1}{ \sin \theta}\right)$

$: \implies \sf \left( \dfrac{ \cos \theta + \sin \theta + 1}{ \cos \theta} \right) \left( \dfrac{ \sin \theta + \cos \theta - 1}{ \sin \theta} \right)$

$: \implies \sf \dfrac{( \cos \theta + \sin \theta) ^{2} - 1 }{ \cos \theta \sin \theta}$

$: \implies \sf \dfrac{\cos ^{2} \theta + { \sin}^{2} \theta + 2 \cos \theta \sin \theta - 1}{ \cos \theta \sin \theta}$

$: \implies \sf \dfrac{1 - 1 + 2\cos \theta \sin \theta}{ \cos \theta \sin \theta}$

$: \implies \sf 2$

Correct option :- C

Question 5 :-

$\bullet\bf \ 1- \cos^2x = \sin ^2 x$

$: \implies \sf \dfrac{1}{\cos^2x}-1$

$: \implies \sf \dfrac{1-\cos^2x}{\cos^2x}$

$: \implies \sf \dfrac{\sin^2x}{\cos^2x}$

$: \implies \sf tan^2x$

Correct option :- D

Answer 2

SOLUTION 1 :

cos 60° sin 30° - cos 30° sin 60°

We know that :

Trick : First remember the values for sin and then just reverse the numbers you will get cos

cos 60° = 1/2

sin 30° = 1/2

cos 30° = √3/2

sin 60° = √3/2

Now, putting the values

• cos 60° sin 30° - cos 30° sin 60°

• 1/2 × 1/2 - √3/2 × √3/2

• (1/2)² - (√3/2)²

• 1/4 - 3/4

• 1 - 3/4

• - 2/4

• - 1/2

Hence, the answer is option A i.e. -1/2

SOLUTION 2 :

If 5x = sec θ and 5/x = tan θ, x² - 1/x² is equal to

We know that :

sec² θ - tan² θ = 1 [from square relations]

Given :

• sec θ = 5x

• tan θ = 5/x

So, putting the values in the relation we get :

$\sf \mapsto {(5x)}^{2} - {( \frac{5}{x}) }^{2} = 1$

Simplifying further

$\sf \mapsto25 {x}^{2} - \frac{25}{ {x}^{2} } = 1$

$\sf \mapsto {x²- \frac{1}{x} }^{2} = \frac{1}{25}$

Hence, the answer is 1/25

SOLUTION 3 :

The value of sin² θ + 1/1 + tan² θ is equal to

1 + tan² θ can be written as sec² θ

Hence,

$\sf \leadsto \sin^{2} \theta + \frac{1}{ {sec}^{2} \theta }$

And we know that 1/sec θ is cos θ

So,

$\sf \leadsto {sin}^{2} \theta + {cos}^{2} \theta$

$\sf \leadsto 1 \: \: \: \: [from \: square \: relations]$

Hence, the correct answer is option B i.e 1

SOLUTION 4 :

(1 + tan θ + sec θ)(1 + cot θ - cosec θ) is equal to

• (1 + tan θ + sec θ)(1 + cot θ - cosec θ)

• 1(1 + cot θ - cosec θ) + tan θ(1 + cot θ - cosec θ) + sec θ(1 + cot θ - cosec θ)

• 1 + cot θ - cosec θ + tan θ + tan θ cot θ - tan θ sec θ + sec θ + cot θ cosec θ - sec θ cosec θ

We know that :

cosec θ = 1/sin θ

tan θ = 1/cot θ and sin θ/cos θ

cot θ = 1/tan θ and cos θ/sin θ

sec θ = 1/cos θ

$\tiny\sf \odot \: 1 + \frac{1}{tan\theta} - \frac{1}{sin\theta} + \frac{1}{cot\theta} + \cancel{tan\theta \times \frac{1}{tan\theta} } - \frac{sin\theta}{cos\theta} \times \frac{1}{cos\theta} + \frac{1}{cos\theta} + \frac{cos\theta}{sin\theta} \times \frac{1}{sin \theta} - \frac{1}{cos \theta} \times \frac{1}{sin\theta}$

$\sf \odot \: \sf \frac{\cos ^{2} \theta + { \sin}^{2} \theta + 2 \cos \theta \sin \theta - 1}{ \cos \theta \sin \theta}$

$\sf \odot \: \sf \frac{1 - 1 + 2\cos \theta \sin \theta}{ \cos \theta \sin \theta}$

1 - 1 and cos θ sin θ/ cos θ sin θ will be cancelled

$\sf \odot \: 2$

Hence, 2 is the answer, correct option is C

SOLUTION 5 :

1/cos² θ - 1 is equal to

1/cos θ is sec θ

• sec² θ - 1

• tan² θ

Hence, correct answer is option D i.e. tan² θ