Question

Sin y +cos y + tan y sin y = sec y +cos y tan y. Verify the Identity. Show all Steps!

Answer 1

`\bf sin(y)+cos(y)+tan(y)sin(y)=sec(y)+cos(y)tan(y) \\\\[-0.35em] ~\dotfill\\\\ sin(y)+cos(y)+tan(y)sin(y)\implies sin(y)+cos(y)+\cfrac{sin(y)}{cos(y)}\cdot sin(y) \\\\\\ sin(y)+cos(y)+\cfrac{sin^2(y)}{cos(y)}\implies \stackrel{\textit{using the LCD of cos(y)}}{\cfrac{sin(y)cos(y)+cos^2(y)+sin^2(y)}{cos(y)}} \\\\\\ \cfrac{sin(y)cos(y)+\stackrel{cos^2(y)+sin^2(y)}{1}}{cos(y)}\implies \cfrac{sin(y)cos(y)+1}{cos(y)} \\\\\\ \cfrac{sin(y)}{cos(y)}\cdot cos(y)+\cfrac{1}{cos(y)}\implies tan(y)cos(y)+sec(y)`