Q&A - Ask Doubts and Get Answers

Sort by :
Clear All
Q

4.     State whether the following are true or false. Justify your answer.

$(i) \sin (A + B) = \sin A + \sin B$
$(ii)$  The value of $\sin \theta$  increases as $\theta$ increases.
$(iii)$ The value of $\cos \theta$ increases as $\theta$ increases.
$(iv)\sin \theta =\cos \theta$  for all values of $\theta$ .
$(v) \cot A$ is not defined for $A=0^{o}$

(i) False, Let A = B = Then,    (ii) True, Take  whent   = 0 then zero(0),   = 30 then value of  is 1/2 = 0.5  = 45 then value of  is 0.707   (iii) False,   (iv) False, Let  = 0    (v) True,   (not defined)

3.     If  $\tan (A+B)=\sqrt{3}$ and $\dpi{100} \tan (A-B)= \frac{1}{\sqrt{3}};$  $0^{o} B,$ find $A \: and \: B.$

Given that, So, ..........(i) therefore, .......(ii) By solving the equation (i) and (ii) we get; and

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

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

We need to prove, Taking LHS; Taking RHS; LHS = RHS Hence proved.

2. Choose the correct option and justify your choice :

$(iv)\frac{2\: \tan 30^{o}}{1-tan^{2}\: 30^{o}}=$

$(A)\: \cos 60^{o}$                $(B)\: \sin 60^{o}$              $(C)\: \tan 60^{o}$            $(D)\: \sin 30^{o}$

Put the value of The correct option is (C)

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

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

[Hint : Simplify LHS and RHS separately]

We need to prove- Taking LHS; Taking RHS; LHS = RHS Hence proved.

2. Choose the correct option and justify your choice :

$(iii)\sin \: 2A=2\: \sin A$ is true when $A$ =

$(A)0^{o}$                $(B)\: 30^{o}$                $(C)\: 45^{o}$                $(D)\: 60^{o}$

The correct option is (A) We know that  So,

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

$(viii)(\sin A+\csc A)^{2}+(\cos A+\sec A)^{2}= 7+\tan ^{2}A+\cot ^{2}A$

Given equation, ..................(i) Taking LHS; [since ] Hence proved

2. Choose the correct option and justify your choice :

$(ii)\: \frac{1-\tan^{2}45^{o}}{1+ \tan^{2}45^{o}}=$

$(A)\: \tan \: 90^{o}$               $(B)\: 1$            $(C) \: \sin 45^{o}$             $(D) \: 0$

The correct option is (D) We know that  So,

2.  Choose the correct option and justify your choice :

$(i)\, \frac{2\: \tan 30^{o}}{1+\tan ^{2}30^{o}}=$

$(A)\: \sin 60^{o}$                $(B)\: \cos 60^{o}$             $(C)\: \tan 60^{o}$            $(D)\: \sin 30^{o}$

Put the value of tan 30 in the given question- The correct option is (A)

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

$(vii)\frac{\sin \theta -2\sin ^{3}\theta }{2\cos ^{3}\theta -\cos \theta }= \tan \theta$

We need to prove - Taking LHS; [we know the identity ] Hence proved.

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

$(vi)\sqrt{\frac{1+\sin A}{1-\sin A}}= \sec A+\tan A$

We need to prove - Taking LHS; By rationalising the denominator, we get; Hence proved.

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

$(v) \frac{\cos A-\sin A+1}{\cos A+\sin A-1}= \csc A+\cot A$ , using the identity $\csc ^{2}A= 1+\cot ^{2}A$

We need to prove - Dividing the numerator and denominator by , we get; Hence Proved.

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

$(iii)\frac{\tan \theta }{1-\cot \theta }+\frac{\cot \theta }{1-\tan \theta }=1+\sec \theta \csc \theta$

[Hint : Write the expression in terms of $\sin \theta$ and $\cos\theta$]

We need to prove- Taking LHS; By using the identity a3 - b3 =(a - b) (a2 + b2+ab) Hence proved.

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

$(ii) \frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}= 2\sec A$

We need to prove- taking LHS;   = RHS Hence proved.

1.        Evaluate the following :

$(v)\frac{5\cos^{2}60^{o}+ 4\sec^{2}30^{o}-\tan^{2}45^{o}}{\sin^{2}30^{o}+\cos^{2}30^{o}}$

.....................(i) We know the values of- By substituting all these values in equation(i), we get;

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

$(i) (\csc \theta -\cot \theta )^{2}= \frac{1-\cos \theta }{1+\cos \theta }$

We need to prove- Now, taking LHS,                                                                        LHS = RHS Hence proved.

1.     Evaluate the following :

$(iv)\: \frac{\sin 30^{o}+\tan 45^{o}-cosec 60^{o}}{\sec 30^{o}+\cos 60^{o}+\cot 45^{o}}$

..................(i) It is known that the values of the given trigonometric functions, Put all these values in equation (i), we get;

1.     Evaluate the following :

$(iii)\: \frac{\cos 45^{o}}{\sec 30^{o}+\csc 30^{o}}$

we know the value of   ,  and , After putting these values

1.     Evaluate the following :

$(ii)\: 2\tan ^{2}45^{o}+ 2\cos ^{2}30^{o}- 2\sin ^{2}60^{o}$

We know the value of   and   According to question,

4.    Choose the correct option. Justify your choice.

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

$(A) \sec ^{2}A$   $(B) -1$  $(C) \cot ^{2}A$    $(D) \tan ^{2}A$

The correct option is (D) ..........................eq (i) The above equation can be written as; We know that  therefore,
Exams
Articles
Questions