Question

tan 70° - tan 20°= k cot 40° k=?​

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

Answer:

The value of k is 2.

Step-by-step explanation:

2sinθcosθ = sin2θ

cos²θ-sin²θ = cos2θ

tan(90-θ) = cotθ

Here we applying the above rules to solve the value of k.

tan70° - tan20° = k cot40°

tan(90°-20°) - tan(20°) = k cot40°

We know that tan(90° - θ) = cotθ

cot20° - tan20° = k cot40°

$\frac{cos20}{sin20} -\frac{sin20}{cos20} = k cot40$

Take LCM(sin20,cos20)

$\frac{cos^{2} 20-sin^{2} 20}{cos20*sin20} = k cot40$

$\frac{cos^{2} 20-sin^{2} 20}{\frac{1}{2}*2 cos20*sin20} = k cot40$  

Here, we applying sin2θ = 2sinθcosθ

$\frac{cos^{2} 20-sin^{2} 20}{\frac{1}{2}*sin2(20)} = k cot40\\\frac{cos^{2} 20-sin^{2} 20}{\frac{1}{2}*sin(40)} = k cot40$

We are applying cos2θ = cos²θ - sin²θ

$\frac{cos2(20)}{\frac{1}{2}*sin(40)} = k cot40\\\frac{cos(40)}{\frac{1}{2}*sin(40)} = k cot40$

$\frac{2cos(40)}{sin(40)} = k \frac{cos40}{sin40}\\\\2 = k$

Therefore, the value of k = 2.