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Standard XII

Mathematics

Principal Solution of Trigonometric Equation

Which of the ...

Question

Which of the following is / are the value (S) of the expression?
sin A(1+ tan A) + cos A (1+ cot A) ?
1. sec A + cosec A
2. 2 cosec A ( sin A + cos A )
3. tan A + cot A
Select the correct answer using the code given below.

1 only

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1 and 2 only

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2 only

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1 and 3 only

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Solution

The correct option is A 1 only
$sin A (1 + tan A) + cos A (1 + cot A)$
= $sin A (1 + \frac{sin A}{cos A}) + cos A (1 + \frac{sin A}{cos A})$
= $(sin A + cos A) (\frac{sin A}{cos A} + \frac{cos A}{sin A})$
= $(sin A + cos A) (\frac{{sin}^{2} A + {cos}^{2} A}{sin A + cos A}$
= $sec A + c o s e c A$
Thus, 1 satisfies the given equation.

Suggest Corrections

Similar questions

Q. If

x = cosec ({tan}^{- 1} (cos ({cot}^{- 1} (sec ({sin}^{- 1} a)))))

and

y = sec ({cot}^{- 1} (sin ({tan}^{- 1} (cosec ({cos}^{- 1} a))))),

where

a \in [0, 1],

then which of the following options is(are) CORRECT?

Tan³ A/ 1+tan²A + cot³A / 1+ cot²A = secA cosecA - 2sinA cosA

Q. Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(i)

{(cosec θ - cot θ)}^{2} = \frac{1 - cos θ}{1 + cos θ}

(ii)

\frac{cos A}{1 + sin A} + \frac{1 + sin A}{cos A} = 2 sec A

(iii)

\frac{tan θ}{1 - cot θ} + \frac{cot θ}{1 - tan θ} = 1 + sec θ cosec θ

[Hint: Write the expression in terms of

sin θ

and

cos θ

(iv)

\frac{1 + sec A}{sec A} = \frac{{sin}^{2} A}{1 - cos A}

[Hint: Simplify LHS and RHS separately].

(v)

\frac{cos A - sin A - 1}{cos A + sin A + 1} = cosec A + cot A

, Using the identity

{cosec}^{2} A = 1 + {cot}^{2} A

(vi)

\sqrt{\frac{1 + sin A}{1 - sin A}} = sec A + tan A

(vii)

\frac{sin θ - 2 {sin}^{3} θ}{2 {cos}^{3} θ - cos θ}

(viii)

{(sin A + cosec A)}^{2} + {(cos A + sec A)}^{2} = 7 + {tan}^{2} A + {cot}^{2} A

(ix)

(cosec A - sin A) (sec A - cos A) = \frac{1}{tan A + cot A}

[Hint: Simplify LHS and RHS separately]

(x)

(\frac{1 + {tan}^{2} A}{1 + {cot}^{2} A}) = {(\frac{1 - tan A}{1 - cot A})}^{2} = {tan}^{2} A

Q. Prove the following trigonometric identities.

(1 + \cot A + \tan A) (\sin A - \cos A) = \frac{\sec A}{{cosec}^{2} A} - \frac{cosec A}{\sec^{2} A} = \sin A \tan A - \cot A \cos A

Q. If

A = π / 6

and

B = π / 3

, then which of the following is/are correct?
1.

sin A + sin B = cos A + cos B

tan A + tan B = cot A + cot B

Select the correct answer using the code given below

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Which of the following is / are the value (S) of the expression?sin A(1+ tan A) + cos A (1+ cot A) ?1. sec A + cosec A2. 2 cosec A ( sin A + cos A )3. tan A + cot A Select the correct answer using the code given below.

The correct option is A 1 onlysinA(1+tanA)+cosA(1+cotA)= sinA(1+sinAcosA)+cosA(1+sinAcosA)= (sinA+cosA)(sinAcosA+cosAsinA)= (sinA+cosA)(sin2A+cos2AsinA+cosA= secA+cosecAThus, 1 satisfies the given equation.

Which of the following is / are the value (S) of the expression?
sin A(1+ tan A) + cos A (1+ cot A) ?
1. sec A + cosec A
2. 2 cosec A ( sin A + cos A )
3. tan A + cot A
Select the correct answer using the code given below.