Question

$\sf \: Solve \: without \: using \: trigonometrical \: tables$

$\color{gold}\large \sf {\left( \frac{tan \: {20}^{ \circ} \: \: }{cosec \: {70}^{ \circ} } \right)}^{2} \: + \: \left( \frac{cot \: {20}^{ \circ} }{sec \: {70}^{ \circ} } \right){}^{2} \: + \: 2 \: tan \: {15}^{ \circ} \: tan \: {45}^{ \circ} \: tan \: {75}^{ \circ}$
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Answer 1

Answer:

Hope you like the answer

Thanks

Answer 2

Answer:

3

Step-by-step explanation:

$\begin{array}{l}(\frac{\tan20^{\circ}}{\csc70^{\circ}})^2+(\frac{\cot20^{\circ}}{\sec70{\circ}})^2+2\tan15^{\circ}.\tan45^{\circ}\tan75^{\circ}\\\;\\=[\frac{\tan(90^{\circ}-70^{\circ})}{\csc70^{\circ}}]^2+[\frac{\cot(90^{\circ}-70^{\circ})}{\sec70^{\circ}}]^2+2\tan(90^{\circ}-75^{\circ}).\tan45^{\circ}\tan75^{\circ}\end{array}$

$\begin{array}{l}=[\frac{\cot70^{\circ}}{\csc70^{\circ}}]^2+[\frac{\tan70^{\circ}}{\sec70^{\circ}}]^2+2\cot75^{\circ}.\tan45^{\circ}.\tan75^{\circ}\end{array}$

Now consider,

$\begin{array}{l}\frac{\cot70^{\circ}}{\csc70^{\circ}}\\\;\\=\frac{\frac{\cos70^{\circ}}{\sin70^{\circ}}}{\frac{1}{\sin70^{\circ}}}\\\;\\=\cos70^{\circ}\end{array}$

and,

$\begin{array}{l}\frac{\tan70^{\circ}}{\sec70^{\circ}}\\\;\\=\frac{\frac{\sin70^{\circ}}{\cos70^{\circ}}}{\frac{1}{\cos70^{\circ}}}\\\;\\=\sin70^{\circ}\end{array}$

Now, put these values in above equation we get,

$\begin{array}{l}[\frac{\cot70^{\circ}}{\csc70^{\circ}}]^2+[\frac{\tan70^{\circ}}{\sec70^{\circ}}]^2+2\cot75^{\circ}.\tan45^{\circ}.\tan75^{\circ}\\\;\\=(\cos70^{\circ})^2+(\sin70^{\circ})^2+2\cot75^{\circ}.\tan45^{\circ}.\tan75^{\circ}\\\;\\=\cos^270^{\circ}+\sin^270^{\circ}+2\cot75^{\circ}.\tan45^{\circ}.\tan75^{\circ}\\\;\\=1+2\cot75^{\circ}.\tan45^{\circ}.\tan75^{\circ}\\\;\\=1+2.\frac{1}{\tan75^{\circ}}.\tan45^{\circ}\tan75^{\circ}\end{array}$

$\begin{array}{l}=1+2\tan45^{\circ}\\\;\\=1+2.1\\\;\\=1+2\\\;\\=3\end{array}$

Hence the required answer is 3.

Note:-

1. tan(90° - Ф) = cotФ

2. cot(90° - Ф) = tanФ

3. sinФ = 1/cosФ

4. cosФ = 1/sinФ

5. tan45° = 1

6. tanФ = sinФ\cosФ

7. cotФ = cosФ/sinФ

8. secФ = 1/cosФ

9. sinФ = 1/cosecФ