Question

The exact value of cosec10 - 4sin70

Answer 1

Cosec10 - 4sin70
= (1/sin10) - [4 sin (90-20)]
=(1/sin10) - (4 cos 20)
= (1/sin10)(1 - 4cos20sin10)
= (1/sin10)[1 - 2(2cos20sin10)]
= (1/sin10)[1 - 2(sin30 - sin10)]
= (1/sin10)[1 - 2(1/2 - sin10)]
= (1/sin10)[1 - 1 + 2sin10]
= (1/sin10)[2sin10]
= 2  Ans.

Answer 2

Answer:

The exact value of the expression is $\csc 10-4\sin 70=2$

Step-by-step explanation:

Given : Expression $\csc 10-4\sin 70$

To find : The exact value of expression ?

Solution :

Expression $\csc 10-4\sin 70$

Re-write as,

$=\frac{1}{\sin 10}-4\sin (90-20)$

$=\frac{1}{\sin 10}-4\cos 20$

$=\frac{1}{\sin 10}(1-4\cos 20\sin10)$

$=\frac{1}{\sin 10}(1-2(2\cos 20\sin10))$

Using formula, $2\cos A\sin B=\sin (A+B)-\sin (A-B)$

$=\frac{1}{\sin 10}(1-2(\sin 30-\sin 10))$

$=\frac{1}{\sin 10}(1-2(\frac{1}{2}-\sin 10))$

$=\frac{1}{\sin 10}(1-1+2\sin 10)$

$=\frac{1}{\sin 10}(2\sin 10)$

$=2$

Therefore, The exact value of the expression is $\csc 10-4\sin 70=2$