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Let `y=2cos^(2)(x/2)sin^(2)x=x^(2)+1/x^(2) rArr y=(1+cosx)sin^(2)x` and `y=x^(2)+1/x^(2)` <br> When `y=(1+cosx)sin^(2)x`= (a number `lt 2`) (a number ` le1`) `rArr y lt 2`……………(i) <br> and when `y=x^(2)+1/x^(2)=(x-1/x)^(2)+2 ge2 rArr y ge 2`………….(ii) <br> No value of y can be obtained satisfying (i) and (ii), simultaneously <br> `rArr` No real solution of the equation exits. **Trigonometric equations and their solutions**

**(i) Prove that general solution of `sintheta=0` is given by `theta=npi; n in Z` (ii) Prove that general solution of `costheta=0` is `theta=((2n+1)pi/2); n in Z`**

**(iii)Prove that general solution of `tan theta=0` is `theta=npi; n in Z` (iv)Prove that the general solution of `cot theta=0` is `theta=(2n+1)pi/2, n in Z`**

**Prove that general solution of `sintheta=sin alpha` is given by `theta=npi+(-1)^n alpha, n in Z`**

**Find the general solution of the equations (i)`sin theta= sqrt3/2` (ii) `2sintheta+1=0` (iii)`cosectheta=2`**

**Prove that the general solution of `costheta=cosalpha`is given by `theta=2npipmalpha`; where `n inZ`**

**Find the general solution of the equation (i)`costheta=1/2 (ii) cos3theta=-1/2 (iii)sqrt3sec2theta=2`**

**Prove that general solution of `tan theta= tan alpha` is given by `theta=npi+alpha;n in Z`**

**Find the general solution of (i) `tan theta=1/sqrt3` (ii) `tan2theta=sqrt3` (iii)`tan3theta=-1`**

**General solution of `(i) sin^2theta = sin^2alpha; (ii) cos^2 theta = cos^2 alpha (iii) tan^2 theta = tan^2 alpha`**