Question

Evaluate :
cos(3π/2+x).cos(2π+X)[cot(3π/2-x)+cot(2π+x)]​

Answer 1

Answer:

sinx.cosx

Step-by-step explanation:

To Find :-

Value of :-

cos(3π/2 + x).cos(2π + x).cot(3π/2 - x).cot(2π + x)

Solution :-

cos(3π/2 + x).cos(2π + x).cot(3π/2 - x).cot(2π + x)

= cos(3(180°)/2 + x).cos(2(180) + x).cot(3(180)/2 - x).cot(2(180) + x)

= cos(270° + x).cos(360° + x).cot(270° - x).cot(360° + x)

= sinx.cosx.tanx.1/tanx

= sinx.cosx.1

= sinx.cosx

cos(3π/2 + x).cos(2π + x).cot(3π/2 - x).cot(2π + x) = sinx.cosx

Steps to remember while writing allied angles :-

1) We need to check that the angle lies in which quadrant.

If :-

90° - A → first quadrant

90° + A → second quadrant

180° - A → second quadrant

180° + A → Third quadrant

270° - A → Third quadrant

270° + A → Fourth quadrant

360° - A → Fourth quadrant

360° + A → First quadrant

2) As we checked the quadrant we need to check wether the ratio is positive or negative in that quadrant :-

1st quadrant :- All ratios are positive

2nd quadrant :- Only sin , cosec ratios are postive

3rd quadrant :- Only tan , cot ratios are Positive

4th quadrant :- Only cos , sec ratios are positive

3) Check wether the angle is an odd multiple or even multiple of an 90°.

4) As , 90° , 270° are odd multiples of 90° in this cases the ratio interchanges linke :-

sin → cos

cos → sin

cosec → sec

sec → cosec

tan → cot

cot → tan

5) As 180° , 360° are even multiples of 90° in this case the ration does not interchanges , it remains same.