sin x tan xcot x csc x sin x tan x sec x csc xtan x cot

sin x +tan x/cot x + csc x = sin x tan x sec x + csc x/tan x + cot x = sin x + cos x sin^3 beta + cos^3 beta/cos beta + sin beta = 1 - sin beta cos beta

Solution

4). convert all the terms into sin, cos

L.H.S= (sinx+tanx) /(cotx+cscx)

= (sinx + sinx/cosx) / ( cosx/sinx + 1/sinx)

= sinx( 1 + 1/cosx) / 1/sinx (cosx+1)

= sin^2 x (1+cosx) /cosx(1+cosx)

= sin^2 x /cosx

= sinx. sinx/cosx

= sinx. tanx

L.H.S= R.H.S

5).

L.H.S= (secx+cscx)/(tanx+cotx)

= (1/cosx +1/sinx) / (sinx/cosx +cosx/sinx)

= [(sinx +cosx)/cosx.sinx] / [ (sin^2 x+cos^2 x)/cosx.sinx]

= (sinx+cosx) / 1

=(sinx + cosx)

L.H.S= R.H.S

6).

a^3 +b^3 = (a+b) (a^2 +b^2 -ab)

apply this formula

L.H.S= (sin^3 + cos^3 )/ (cos+sin)

= [ (sin+cos) (sin^2+cos^2 -sincos)] /(cos+sin)

= (sin^2 +cos^2 -sincos)

= (1 - sin.cos)

L.H.S= R.H.S