Given sinA = 3/4, find cosA cosecA + tanA secA

Given that \(\sin A = \frac{3}{4}\), we need to find the value of \(\cos A \csc A + \tan A \sec A\).

### Step 1: Find \(\cos A\)
We know:
\[
\sin^2 A + \cos^2 A = 1
\]
Substituting \(\sin A = \frac{3}{4}\):
\[
\left(\frac{3}{4}\right)^2 + \cos^2 A = 1
\]
\[
\frac{9}{16} + \cos^2 A = 1
\]
\[
\cos^2 A = 1 - \frac{9}{16} = \frac{7}{16}
\]
\[
\cos A = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4}
\]

### Step 2: Find \(\csc A\)
\[
\csc A = \frac{1}{\sin A} = \frac{1}{\frac{3}{4}} = \frac{4}{3}
\]

### Step 3: Find \(\tan A\)
\[
\tan A = \frac{\sin A}{\cos A} = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}} = \frac{3\sqrt{7}}{7}
\]

### Step 4: Find \(\sec A\)
\[
\sec A = \frac{1}{\cos A} = \frac{1}{\frac{\sqrt{7}}{4}} = \frac{4}{\sqrt{7}} = \frac{4\sqrt{7}}{7}
\]

### Step 5: Calculate \(\cos A \csc A + \tan A \sec A\)
Now we can substitute the values:
\[
\cos A \csc A = \frac{\sqrt{7}}{4} \times \frac{4}{3} = \frac{\sqrt{7}}{3}
\]
\[
\tan A \sec A = \frac{3\sqrt{7}}{7} \times \frac{4\sqrt{7}}{7} = \frac{12 \times 7}{49} = \frac{84}{49} = \frac{12}{7}
\]
Adding these together:
\[
\cos A \csc A + \tan A \sec A = \frac{\sqrt{7}}{3} + \frac{12}{

Learn how to solve the trigonometric problem: Given sinA = 3/4, find cosA cosecA + tanA secA. This involves applying trigonometric identities and solving step-by-step to find the required value.

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