Question

Q18)
prove that : sin (n+1) x sin (n+2)x + cos (n+1) x cos (n+2)x =cosx

18. Prove that : sin ( n + 1 ) x sin ( n + 2 ) x + cos ( n + 1 ) x cos ( n+ 2) x = cosx

Answer 1

Dear student,

$$\begin{gathered} \mathrm{LHS}=\sin \left(\mathrm{n}+1\right)\mathrm{x}.\sin \left(\mathrm{n}+2\right)\mathrm{x}+\cos \left(\mathrm{n}+1\right)\mathrm{x}.\cos \left(\mathrm{n}+2\right)\mathrm{x} \\ =\cos \left[\left(\mathrm{n}+2\right)\mathrm{x}-\left(\mathrm{n}+1\right)\mathrm{x}\right] \\ =\cos \left(\mathrm{nx}+2\mathrm{x}-\mathrm{nx}-\mathrm{x}\right) \\ =\cos \mathrm{x} \\ =\mathrm{RHS} \\ \boxed{\mathbf{Using}\mathbf{}\mathbf{identity}\mathbf{}\mathbf{:} \\ \mathbf{cos}\mathbf{}\mathbf{A}\mathbf{}\mathbf{.}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{B}\mathbf{}\mathbf{+}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{A}\mathbf{}\mathbf{.}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{cos}\left(\mathbf{A}\mathbf{}\mathbf{-}\mathbf{}\mathbf{B}\right)} \end{gathered}$$

Hence Proved.

Regards