Find the amplitude of `sinalpha+i(1-cosalpha)`

If alpha is in first quadrant such that tan^(2)alpha=(8)/(7) , then the value of ((1+sinalpha)(1-sinalpha))/((1+cosalpha)(1-cosalpha))

If sinbeta is the harmonic mean of sinalpha and cosalpha , and sintheta is the arithmetic mean of sinalpha" and "cosalpha , then which of the following is/are correct? 1. sqrt(2)sin(alpha+(pi)/(4))sinbeta=sin2alpha 2. sqrt(2)sintheta=cos(alpha-(pi)/(4)) Select the correct answer using the code given below:

If 0^(@)ltalphalt90^(@) , then solve the equation (sinalpha)/(1-cosalpha)+(sinalpha)/(1+cosalpha)=4 .

If 7sinalpha=24cosalpha,0ltalphalt(pi)/(2) then the value of 14tanalpha-75cosalpha-7secalpha is equal to

Let lamda and alpha real. Find the set of all values of lamda for which the system. x+(sinalpha)y+(cosalpha)z=0, x+(cosalpha)y+(sinalpha)z=0, -x+(sinalpha)y+(cosalpha)z=0 has a non trivial solution. For lamda=1 find all values of alpa.

If tantheta=(sinalpha-cosalpha)/(sinalpha+cosalpha) , then sinalpha+cosalphaandsinalpha-cosalpha are equal to

Show without expanding at any stage that: [0,sinalpha-cosalpha],[-sinalpha,0,sinbeta],[cosalphas-sinbeta,0]|=0

Show without expanding at any stage that: | (1,cosalpha-sinalpha, cosalpha+sinalpha),(1,cosbeta-sinbeta,cosbeta+sinbeta),(1, cosgamma-singamma,cosgamma+singamma)| =2 |(1,cosalpha, sinalpha),(1,cosbeta, sinbeta),(1,cosgamma,singamma)|

If (2sinalpha)/(1+cosalpha +sinalpha)=y , then prove that (1-cosalpha+sinalpha )/(1+sinalpha) is also equal to y.