Question

5 Evaluate the following cosctan (D) 6 Simplify the following Seix cos x usines cosy) > Verify the following identity csc@stanx= secx csak

Please solve all equations with each stop listed along with a brief explanation if needed. I am looking for both the steps and solution, thanks. I’ll give a quick thumbs up for great answers.

Please solve all equations with each stop listed along with a brief explanation if needed. I am looking for both the steps and solution, thanks. I’ll give a quick thumbs up for great answers.

Answer 1

Solution

5)cos(arctan7/9)

let x=arctan(7/9)

means to find cosx=?

then tanx=7/9

now, tanx=sinx/cosx so,

sinx/cosx=7/9

squaring both sides we get

(Sinx)^2/(cosx)^2=49/81

adding 1 to both sides then we get

[(Sinx)^2/(cosx)^2]+1=(49/81)+1

[(Sinx)^2+(cosx)^2]/(cosx)^2=130/81

Since, (sinx)^2+(cosx)^2=1

so, 1/(cosx)^2=130/81

​​​​​​and (cosx)^2=81/130

and cosx=√(81/130)

Cosx=9/√130

so, cos(arctan7/9)=9/√130

6)secx[(cosx)^3+(sinx)^2 cosx]

Since, (sinx)^2+(cosx)^2=1

so, (sinx)^2=1-(cosx)^2

Secx[(cosx)^3+[1-(cosx)^2]cosx]

secx[(cosx)^3+cosx-(cosx)^3]

Note:-(cosx)^3-(cosx)^3=0

so, we get

secx[cosx]

Since, secx=1/cosx

(1/cosx)×cosx

now cosx and cosx will cancel out and we get

1

​​​​​​7)(cscx)^2tanx=secxcscx

left hand side

cscx=1/sinx and tanx=sinx/cosx

so, (1/(sinx)^2)×sinx/cosx

sinx/(sinx)^2×cosx

[1/(sinx)×(cosx)]

now taking right hand side

secx × cscx

since, secx=1/cosx and cscx=1/sinx

so, (1/cosx)×(1/sinx)

1/(cosx)×(sinx)

so, left hand side=right hand side

hence proved.