Simplify the expression [1 - sin² 32° + 1 / (1 + tan² 58°)].

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1747981786/aptidude_questions/x0ztv2cdyvfa01qln53r.png)

Question

Simplify the expression [1 - sin² 32° + 1 / (1 + tan² 58°)].

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1747981786/aptidude_questions/x0ztv2cdyvfa01qln53r.png)

Accepted Answer

Step 1: Use the Pythagorean identity: 1 - sin² A = cos² A  
For A = 32°, 1 - sin² 32° = cos² 32°

Step 2: Use the identity 1 + tan² A = sec² A  
For A = 58°, 1 + tan² 58° = sec² 58°  
Therefore, 1 / (1 + tan² 58°) = 1 / sec² 58° = cos² 58°

Step 3: Now the expression becomes:  
cos² 32° + cos² 58°

Step 4: Note that 58° = 90° - 32°, so cos 58° = sin 32°  
Therefore, cos² 58° = sin² 32°

Step 5: Substitute back:  
cos² 32° + sin² 32° = 1 (Pythagorean identity)

Hence, the simplified value is 1.

Answer

Answer

Answer