`alpha,beta,gammaa n ddelta` are angle in I, II, III and iv quadrants, respectively and none of them is an integral multiple of `pi/2dot` They form an increasing arithmetic progression. Which of the following holds? `cos(alpha+delta)>0` `cos(alpha+delta)=0` `cos(alpha+delta)<0` `cos(alpha+delta)>0orcos(alpha+delta)<0` Which of the following does not hold? `sin(beta+gamma)=sin(alpha+delta)` `sin(beta-gamma)="sin"(alpha-delta)` `sin(alpha-beta)="tan"(beta-delta)` `sin(alpha+gamma)=cos2beta)` If `alpha+beta+gamma+delta=thetaa n dalpha=70^0,` then `400^0

Options

A. `cos(alpha-delta)gt0`
B. `cos(alpha-delta)=0`
C. `cos(alpha-delta)lt0`
D. `cos(alpha-delta)gt0or cos(alpha-delta)lt0`

Answer

A

Explanation

`0ltalphalt90^@,90^@ltbetalt180^@`
`180^@ltgammalt270^@,270^@ltdeltalt360^@`
`rArrgammalt270^@ltalpha+deltalt450^@`
`rArr alpha+delta` lies in the I or IV quadrant and cosine in both is positive.
If d is the common ratio of theA.P., then
`beta=alpha+d,gamma=alpha+2d,delta=alpha+3d`
`rArr beta+gamma=alpha+delta,2(alpha-beta)=-2d=beta-delta`
`and alpha+gamma=2beta`,
Now, `beta-gamma=-d,alpha-delta=-3d`
`270^@ltdeltalt360^@`
`rArr 270^@ltalpha+3dlt360^@`
`rArr 200^@lt3dlt290^@ ( :. alpha=70^@)`
`rArr400^@lt+6d,580^@`
`rArr 680^@lt4alpha+6dlt860^@`
`680^@ lttheta lt 860^@`