(a) If cosθ=53, find the value of cosec θ−cotθtanθ+sinθ.
OR
(b) For θ=30∘, verify that cos2θ=cos2θ−sin2θ.
Get the complete, step-by-step math solution for: "(a) If cos θ = (3)/(5), find the value of {tan θ + sin θ}{cosec θ - cot θ}. OR (b) For θ = 30^ , verify that cos 2θ = cos² θ - sin² θ.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Calculate sinθ and tanθ
Given cosθ=53, we can use the identity sin2θ+cos2θ=1 to find sinθ. Once sinθ is known, we can calculate tanθ using the identity tanθ=cosθsinθ.
Step 2: Calculate cosec θ and cotθ
Next, we find the reciprocal trigonometric ratios. cosec θ is the reciprocal of sinθ, and cotθ is the reciprocal of tanθ.
Step 3: Substitute values into the expression
Now we substitute the calculated values of sinθ, tanθ, cosec θ, and cotθ into the given expression.
Step 4: Simplify the expression
We simplify the numerator and the denominator separately. The numerator becomes 1520+12=1532, and the denominator becomes 42=21. Finally, we divide the numerator by the denominator.