Question

prove that sin34°/sin56°+cos34°/cos56°=sec34° cosec34°

Answer 1

I hope this will help u

Answer 2

Answer:  Proved.

Step-by-step explanation: We are given to prove the following equality -

$\dfrac{\sin 34^{\circ}}{\sin 56^{\circ}}+\dfrac{\cos 34^{\circ}}{\cos 56^{\circ}}=\sec 34^{\circ} \csc 34^{\circ} .$

To prove the above, either we can start from right-hand side and arrive at left-hand side or we can start from left-hand side and prove that it is equal to right-hand side.

Let us begin with left-hand side and will try to arrive at the right-hand side.

$L.H.S=\dfrac{\sin 34^{\circ}}{\sin 56^{\circ}}+\dfrac{\cos 34^{\circ}}{\cos 56^{\circ}}\\ \Rightarrow L.H.S=\dfrac{\sin 34^{\circ}\cos 56^{\circ}+\cos 34^{\circ}\sin 56^{\circ}}{\sin 56^{\circ}\cos 56^{\circ}}\\\Rightarrow L.H.S=\dfrac{\sin(34^{\circ}+56^{\circ})}{\sin 56^{\circ}\cos 56^{\circ}}\\\Rightarrow L.H.S=\dfrac{\sin 90^{\circ}}{\sin 56^{\circ}\cos 56^{\circ}}\\\Rightarrow L.H.S=\dfrac{1}{\sin 56^{\circ}\cos 56^{\circ}}\\\Rightarrow L.H.S=\csc 56^{\circ} \sec 56^{\circ}\\\Rightarrow L.H.S=\csc (90^{\circ}-56^{\circ}) \sec (90^{\circ}-56^{\circ})\\\Rightarrow L.H.S=\sec 34^{\circ} \csc 34^{\circ}\\\Rightarrow L.H.S=R.H.S.$

Hence proved.