Find the simplified form of cos−1(35 cos x+45 sin x) where x∈[−3π4,π4].
We have, cos−1(35 cos x+45 sin x) where x∈[−3π4,π4]
Let cos y=35
⇒ sin y=45⇒ y=cos−135=sin−145=tan−1(43)∴ cos−1[cos y.cos x+sin y. sin x]=cos−1[cos(y−x)] [∵ cos(A−B)=cos A.cos B+sin A.sinB]=y−x=tan−143−x [∵ y=tan−143]