Question

Without using trigonometric tables, find the value of the following:
Cot theta.tan(90°-theta)-sec(90°-theta)cosec theta + under root 3.tan12°.tan 60°.tan 78°

Answer 1

the answer is 2 hope it helps

Answer 2

Answer:

$\cot( \theta) \tan(90 \degree -\theta ) - \sec( 90 \degree -\theta) \cosec( \theta) + \sqrt{3} \tan(12 \degree ) \tan(60 \degree) \tan(78\degree) = 2$

Step-by-step explanation:

1) Step 1:

Simplying Using

tan(90-x) = cotx

sec(90-x) = cosecx

$= \cot( \theta) \cot( \theta ) - \cosec( \theta ) \cosec( \theta ) \: + \sqrt{3} \tan(60 \degree) \tan(12\degree) \tan(78\degree)$

2) Step 2:

Putting the value tan60° = √3

${ \cot^{2} ( \theta) } - { \cosec^{2} ( \theta) } + \sqrt{3} \sqrt{3} \tan(12\degree) \tan(78\degree)$

3) Step 3:

Using Identity

cosec²x - cot²x = 1

So cot²x - cosec²x = -1

$= -1 + 3 \tan(12\degree) \tan(78\degree)$

4) Step 4:

Substituting tan(12) = tan(90-78)

$= -1 + 3 \tan((90-78)\degree) \tan(78\degree)$

5) Step 5:

Again using Identity

tan(90-x) = cotx

and cotx tanx = 1

$= -1 + 3 \cot(78\degree) \tan(78\degree)$

$= -1 + 3(1)$

$= -1 + 3$

$= 2$

So on simplifying the answer comes out to be 2

$\cot( \theta) \tan(90 \degree -\theta ) - \sec( 90 \degree -\theta) \cosec( \theta) + \sqrt{3} \tan(12 \degree ) \tan(60 \degree) \tan(78\degree) = 2$