The value of \frac{\sin 23°\cos 67°+\sec 52°\sin 38°+\cos 23°\sin 67°+\csc 52°\cos 38°}{{\csc }^{2}20°-{\tan }^{2}70°}

The correct answer is 3.

To find the value of the given expression, we can simplify the numerator and the denominator separately using fundamental trigonometric identities.

Step 1: Simplify the Numerator

The numerator is given by:

\sin 23°\cos 67°+\sec 52°\sin 38°+\cos 23°\sin 67°+\csc 52°\cos 38°

We can group and simplify these terms as follows:

1. Grouping the terms involving 23° and 67°:

\sin 23°\cos 67°+\cos 23°\sin 67°

Using the identity \sin (A+B)=\sin A\cos B+\cos A\sin B, we have:

\sin 23°\cos 67°+\cos 23°\sin 67°=\sin (23°+67°)=\sin 90°=1

2. Simplifying the term \sec 52°\sin 38°:

Using the complementary angle identity \sin (90°-\theta )=\cos \theta:

\sin 38°=\sin (90°-52°)=\cos 52°

Therefore:

\sec 52°\sin 38°=\sec 52°\cos 52°=1

3. Simplifying the term \csc 52°\cos 38°:

Using the complementary angle identity \cos (90°-\theta )=\sin \theta:

\cos 38°=\cos (90°-52°)=\sin 52°

Therefore:

\csc 52°\cos 38°=\csc 52°\sin 52°=1

Adding the three simplified components, the numerator becomes:

\text{Numerator}=1+1+1=3

Step 2: Simplify the Denominator

The denominator is given by:

{\csc }^{2}20°-{\tan }^{2}70°

Using the complementary angle identity \tan (90°-\theta )=\cot \theta:

\tan 70°=\tan (90°-20°)=\cot 20°

Substituting this back into the denominator:

{\csc }^{2}20°-{\cot }^{2}20°

Using the standard Pythagorean identity {\csc }^2\theta -{\cot }^2\theta =1:

\text{Denominator}=1

Step 3: Calculate the Final Value

Dividing the numerator by the denominator, we get:

\text{Value}=\frac{3}{1}=3