Question

Without using trigonometric tables, prove that :
`(i) sin 43^@cos47^@+cos43^@ sin 47^@=1`
`(ii) cos 38^@cos 52^@-sin 52^@=0`
`(iii) sec 50^@sin40^@+cos40^@cosec 50^@=2`
`(iv) sec 70 ^@sin 20^@-cos 20^@-cosec 70^@=0`

Answer 1

We have
`(i) sin 43^@cos47^@+cos43^@ sin 47^@`
`=sin (90^@-47^@)cos 47^@+cos (90^@-47^@)sin 47^@`
`=cos^2 47^@ +sin^2 47^@=1`
`[because sin (90-theta)=cos theta, cos (90^@=theta) =sin theta]`.
`(ii) cos 38^@cos 52^@-sin 52^@`
`=cos (90^@-52^@)cos 52^@-sin(90^@-52^@)sin 52^@`
`=sin 52^@=cos 52^@-sin 53^@cos 52^@=0`
`[because cos (90^@-theta) =sin theta and sin (90^@-theta) =cos theta]`.
`(iii) sec 50^@sin40^@+cos40^@cosec 50^@`
`sec (90^@-40^@)sin 40^@+cos 40^@cosec(90^@-40^@)`
`=cosec 40^@sin 40^@+cos 40^@sec 40^@`
`[{:(because sec(90^@-theta) =cosec theta),(cosec(90^@-theta)=sec theta):}]`
` (sin 40^@)/(sin 40^@)+(cos 40^@)/(cos 40^@)=1+1=2`.
`(iv) sec 70 ^@sin 20^@-cos 20^@-cosec 70^@`
`=sec (90^@-20^@)sin 20^@-cos 20^@cosec (90^@-20^@)`
`=cosec 20^@sin 20^@sin 20^@-cos 20^@sec 20^@`
`[{:(because sec(90^@-theta) ="cosec" theta","),("cosec" (90^@-theta)=sec theta):}]`
` (sin20^@)/(sin 20^@)-(cos20^@)/(cos 20^@)=1-1=0`.