Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the third quadrant.

The exact value of <math><mstyle displaystyle="true"><mi>sec</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Evaluate the exponent.

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

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