Question

solve kr do koii...​

Answer 1

Given:-

BC= 15cm

$\rm{sin\ B=\dfrac{4}{5}}$

To Find:-

i) Measurement of AB and AC

ii) Measurement of CD and AD, if tan(∠ADC)=1

To Prove:-

$\longrightarrow\bf{tan^{2}\ B-\dfrac{1}{cos^{2}\ B}=-1}$

Solution:-

In ΔABC,

$\longrightarrow\rm{cos\ B=\sqrt{1-sin^{2}B}}$

$\longrightarrow\rm{cos\ B=\sqrt{1-\bigg(\dfrac{4}{5}\bigg)^{2}}}$

$\longrightarrow\rm{cos\ B=\sqrt{1-\dfrac{16}{25}}}$

$\longrightarrow\rm{cos\ B=\sqrt{\dfrac{25-16}{25}}}$

$\longrightarrow\rm{cos\ B=\sqrt{\dfrac{9}{25}}}$

$\longrightarrow\rm{cos\ B=\dfrac{3}{5}}$

Now,

$\longrightarrow\rm{cos\ B=\dfrac{BC}{AB}}$

$\longrightarrow\rm{\dfrac{3}{5}=\dfrac{15}{AB}}$

$\longrightarrow\rm{AB=\dfrac{15\times5}{3}}$

$\longrightarrow\rm\green{AB=25\ cm}$

Now,

$\longrightarrow\rm{sin\ B=\dfrac{AC}{AB}}$

$\longrightarrow\rm{\dfrac{4}{5}=\dfrac{AC}{25}}$

$\longrightarrow\rm{\dfrac{4\times25}{5}=AC}$

$\longrightarrow\rm{AC=\dfrac{4\times25}{5}}$

$\longrightarrow\rm\green{AC=20\ cm}$

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Now, In ΔACD

tan(∠ADC)=1

That means,

∠ADC= 45°

Or

D= 45°

So,

$\longrightarrow\rm{tan\ D=\dfrac{AC}{CD}}$

$\longrightarrow\rm{tan\ 45^{\circ}=\dfrac{20}{CD}}$

$\longrightarrow\rm{1=\dfrac{20}{CD}}$

$\longrightarrow\rm\green{CD=20\ cm}$

Now,

$\longrightarrow\rm{sin\ D=\dfrac{AC}{AD}}$

$\longrightarrow\rm{sin\ 45^{\circ}=\dfrac{20}{AD}}$

$\longrightarrow\rm{\dfrac{1}{\sqrt{2}} =\dfrac{20}{AD}}$

$\longrightarrow\rm\green{AD=20\sqrt{2}\ cm}$

━━━━━━━━━━━━━━━━━━━━━

Now, we know that,

$\pink{\underline{\boxed{\bf{1+tan^{2}\theta=sec^{2}\theta}}}}$

$\purple{\underline{\boxed{\bf{sec \theta}=\dfrac{1}{cos\theta}}}}$

So, by using above properties

$\longrightarrow\rm{1+tan^{2}B=sec^{2}B}$

$\longrightarrow\rm{tan^{2}B-sec^{2}B=-1}$

$\longrightarrow\rm{tan^{2}B-\bigg(\dfrac{1}{cosB} \bigg)^{2}=-1}$

$\longrightarrow\rm{tan^{2}B-\dfrac{1}{cos^{2}B}=-1}$

$\bigstar\bf\red{Hence\ Proved}$

Answer 2

Photosensitivity is an extreme sensitivity to ultraviolet (UV) rays from the sun and other light sources. Most people are at risk of developing sunburn during long exposure to sunlight. Exposure to UV rays can also lead to skin damage and skin cancer.