We start with the equation 8‌cos‌θ+15sin‌θ=15. We need to find 15‌cos‌θ−8sin‌θ, which we will denote as t. First, consider squaring both sides of the given expression for clarity: Start with (8‌cos‌θ+15sin‌θ)2=152 : 64cos2θ+225sin‌2θ+2⋅8⋅15‌cos‌θ‌s‌i‌n‌θ=225. Simplify to: 64cos2θ+225sin‌2θ+240‌cos‌θ‌s‌i‌n‌θ=225. For t=15‌cos‌θ−8sin‌θ, square both sides: (15‌cos‌θ−8sin‌θ)2=t2. Expanding this gives: 225cos2θ+64sin‌2θ−240‌cos‌θ‌s‌i‌n‌θ=t2. Now, add the two squared expressions: 64cos2θ+225sin‌2θ+240‌cos‌θ‌s‌i‌n‌θ+225cos2θ+64sin‌2θ−240‌cos‌θ‌s‌i‌n‌θ=225 Simplify the combined expression: 289(cos2θ+sin‌2θ)=225+t2. Since cos2θ+sin‌2θ=1, it follows that: 289=225+t2. Solve for t2 : ‌289−225=t2 ‌64=t2 ‌t=8 Thus, 15‌cos‌θ−8sin‌θ=8.