1sec beta tan beta sec beta tan beta 1 cos x1 cos x co

1/sec beta + tan beta = sec beta - tan beta 1 - cos x/1 + cos x = (cot x - csc x)^2 tan alpha - csc alpha sec alpha (1 - 2 cos^2 alpha) = cot alpha 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

1).

L.H.S= 1/(sec+tan)

mutiply and divide with (sec-tan)

=   (sec-tan) / [ (sec+tan).(sec-tan)]

= (sec-tan) / [ sec^2 - tan^2 ]

but sec^2 - tan^2 =1 , so

= (sec - tan)/1

= sec - tan

L.H.S= R.H.S

2).

L.H.S= (1-cosx)/(1+cosx)

mutiply and divide with (1-cosx)

so lhs = (1-cosx)^2 / (1-cos^2 x)

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

= [ (1-cosx)/sinx ] ^2

= [ 1/sinx - cosx/sinx ]^2

= [ cscx - cotx]^2

= (-1)^2 (cotx - cscx)^2

= (cotx - cscx)^2

L.H.S= R.H.S

3).

L.H.S= tan -csc.sec(1- 2cos^2 )

= tan - 1/sinx.cosx ( 1 - 2cos^2)

= sin/cos - (sin^2 +cos^2-2cos^2) / sin.cos

= sin/cos - (sin^2 -cos^2)/sin.cos

= [sin^2 -sin^2 +cos^2] / sin.cos

=[cos^2/sin.cos]

= [cos/sin]

=cot

L.H.S= R.H.S

post other 3 questions in separate question