## Simplify $\left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)$

Updated on 10-Oct-2022 13:29:22
To do:We have to simplify $\left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)$.Solution:We know that,$\sin^2 \theta+\cos ^{2} \theta=1$.....(i)$\sec^2 \theta-\tan ^{2} \theta=1$.......(ii)$\sec \theta \times \cos \theta=1$.......(iii)Therefore,$\left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)=\left(1+\tan ^{2} \theta\right)(1^2-\sin^2 \theta)$          [Since $(a-b)(a+b)=a^2-b^2$]$=\left(\sec ^{2} \theta\right)(\cos^2 \theta)$           [From (i) and (ii)]$=(\sec \theta\times cos \theta)^2$                  $=1^2$                          [From (iii)]$=1$ Therefore,$\left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)=1$.

## If $\sqrt{3} \tan \theta=1$, then find the value of $\sin ^{2} \theta-\cos ^{2} \theta$.

Updated on 10-Oct-2022 13:29:22

## Prove the following:$(\sin \alpha+\cos \alpha)(\tan \alpha+\cot \alpha)=\sec \alpha+\operatorname{cosec} \alpha$

Updated on 10-Oct-2022 13:29:22
Given:$(\sin \alpha+\cos \alpha)(\tan \alpha+\cot \alpha)=\sec \alpha+\operatorname{cosec} \alpha$ To do:We have to prove that $(\sin \alpha+\cos \alpha)(\tan \alpha+\cot \alpha)=\sec \alpha+\operatorname{cosec} \alpha$.Solution:We know that, $\tan \theta =\frac{\sin \theta}{\cos \theta}$$\cot \theta=\frac{\cos \theta}{\sin \theta}$$\sin ^{2} \theta+\cos ^{2} \theta=1$$\sec \theta =\frac{1}{\cos \theta}$$\operatorname{cosec} \theta=\frac{1}{\sin \theta}$Therefore, LHS $=(\sin \alpha+\cos \alpha)(\tan \alpha+\cot \alpha)$$=(\sin \alpha+\cos \alpha)(\frac{\sin \alpha}{\cos \alpha}+\frac{\cos \alpha}{\sin \alpha})$$=(\sin \alpha+\cos \alpha)(\frac{\sin^2 \alpha+\cos^2 \alpha}{\sin \alpha \cos \alpha})$$=(\sin \alpha+\cos \alpha) \frac{1}{(\sin \alpha \cos \alpha)}$$=\frac{1}{\cos \alpha}+\frac{1}{\sin \alpha}$$=\sec \alpha+\operatorname{cosec} \alpha$$=$ RHSHence proved.Read More

## Prove the following:If $\tan \mathrm{A}=\frac{3}{4}$, then $\sin \mathrm{A} \cos \mathrm{A}=\frac{12}{25}$

Updated on 10-Oct-2022 13:29:22

## Prove the following:$\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=2 \operatorname{cosec} \theta$

Updated on 10-Oct-2022 13:29:22
Given:$\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=2 \operatorname{cosec} \theta$To do:We have to prove that $\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=2 \operatorname{cosec} \theta$Solution:We know that, $(a+b)^2=a^2+2ab+b^2$$\sin ^{2} \theta+\cos ^{2} \theta=1Therefore, LHS =\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}$$=\frac{\sin ^{2} \theta+(1+\cos \theta)^{2}}{\sin \theta(1+\cos \theta)}$$=\frac{\sin ^{2} \theta+1+\cos ^{2} \theta+2 \cos \theta}{\sin \theta(1+\cos \theta)}$$=\frac{1+1+2 \cos \theta}{\sin \theta(1+\cos \theta)}$$=\frac{2(1+\cos \theta)}{\sin \theta(1+\cos \theta)}$$=\frac{2}{\sin \theta}$$=2 \operatorname{cosec} \theta$$=$ RHSHence proved.Read More

## Explain the given reactions with the examples(a) Hydrogenation reaction(b) Oxidation reaction(c) Substitution reaction(d) Saponification reaction(e) Combustion reaction

Updated on 10-Oct-2022 13:29:14
(a) Hydrogenation reaction can be defined as an addition reaction between the hydrogen and other compounds in the presence of a catalyst.$CH_2 = CH_2 → CH_3CH_3$(b) An oxidation reaction is a reaction in which oxygen combines with an element or a compound. In other words, it can also be defined as a molecule loses its electrons.$2Mg + O_2→ 2MgO$(c) Substitution reaction can be defined as the reaction in which a more reactive element displaces a less reactive element from its aqueous salt solution.$Zn + CuSO_4 → ZnSO_4 + Cu$(d) Saponification reaction is the reaction in which the hydrolysis of an ... Read More

## Look at Figure 4.1 and answer the following questions(a) What change would you observe in the calcium hydroxide solution taken in tube B?(b) Write the reaction involved in test tubes A and B respectively.(c) If ethanol is given instead of ethanoic acid

Updated on 10-Oct-2022 13:29:14
(a) The changes observed in B is that Calcium Hydroxide solution would become milky.(b) Reaction in Tube A: $CH_3COOH+NaHCO_3 → CH_3COONa + CO_2 + H_2O$Reaction in tube B$Ca(OH)_2 + CO_2 → CaCO_3 + H_2O$(c) If ethanol is replaced with ethanoic acid,  the same change will not be observed as Ethanol will not react with Sodium Hydrogen carbonate.(d) A solution of lime water can be prepared in the laboratory by taking distilled water in a beaker and mixing Calcium carbonate powder to it, thoroughly. Allow the solution to settle and decant the clear solution to obtain the lime water.Read More